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Periods and Nori Motives

Annette Huber und Stefan Muller-Stach,with contributions by Benjamin Friedrich, Jonas von Wangenheim

September 22, 2015

2

Contents

Introduction 9

0 Introduction 11

0.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . 11

0.2 Recent developments . . . . . . . . . . . . . . . . . . . . . . . . . 17

0.3 Nori motives and Tannaka duality . . . . . . . . . . . . . . . . . 18

0.4 Cohomology theories . . . . . . . . . . . . . . . . . . . . . . . . . 20

0.5 Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

0.6 Leitfaden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

0.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . 22

I Background Material 25

1 General Set-up 27

1.1 Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.1.1 Linearizing the category of varieties . . . . . . . . . . . . 27

1.1.2 Divisors with normal crossings . . . . . . . . . . . . . . . 28

1.2 Complex analytic spaces . . . . . . . . . . . . . . . . . . . . . . . 28

1.2.1 Analytification . . . . . . . . . . . . . . . . . . . . . . . . 29

1.3 Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.3.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . 30

1.3.2 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.3.3 Total complexes and signs . . . . . . . . . . . . . . . . . . 31

1.4 Hypercohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3

4 CONTENTS

1.4.2 Godement resolutions . . . . . . . . . . . . . . . . . . . . 34

1.4.3 Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . 36

1.5 Simplicial objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.6 Grothendieck topologies . . . . . . . . . . . . . . . . . . . . . . . 42

1.7 Torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.7.1 Sheaf theoretic definition . . . . . . . . . . . . . . . . . . 45

1.7.2 Torsors in the category of sets . . . . . . . . . . . . . . . 46

1.7.3 Torsors in the category of schemes (without groups) . . . 49

2 Singular Cohomology 53

2.1 Sheaf cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.2 Singular (co)homology . . . . . . . . . . . . . . . . . . . . . . . . 56

2.3 Simplicial cohomology . . . . . . . . . . . . . . . . . . . . . . . . 57

2.4 Kunneth formula and Poincare duality . . . . . . . . . . . . . . . 60

2.5 Basic Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.5.1 Direct proof of Basic Lemma I . . . . . . . . . . . . . . . 66

2.5.2 Nori’s proof of Basic Lemma II . . . . . . . . . . . . . . . 68

2.5.3 Beilinson’s proof of Basic Lemma II . . . . . . . . . . . . 70

2.6 Triangulation of Algebraic Varieties . . . . . . . . . . . . . . . . 75

2.6.1 Semi-algebraic Sets . . . . . . . . . . . . . . . . . . . . . . 75

2.6.2 Semi-algebraic singular chains . . . . . . . . . . . . . . . . 80

2.7 Singular cohomology via the h′-topology . . . . . . . . . . . . . . 84

3 Algebraic de Rham cohomology 87

3.1 The smooth case . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.1.2 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.1.3 Cup product . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.1.4 Change of base field . . . . . . . . . . . . . . . . . . . . . 92

3.1.5 Etale topology . . . . . . . . . . . . . . . . . . . . . . . . 93

3.1.6 Differentials with log poles . . . . . . . . . . . . . . . . . 94

3.2 The general case: via the h-topology . . . . . . . . . . . . . . . . 96

3.3 The general case: alternative approaches . . . . . . . . . . . . . . 100

3.3.1 Deligne’s method . . . . . . . . . . . . . . . . . . . . . . . 100

CONTENTS 5

3.3.2 Hypercovers . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.3.3 Definition of de Rham cohomology in the general case . . 102

3.3.4 Hartshorne’s method . . . . . . . . . . . . . . . . . . . . . 103

3.3.5 Using geometric motives . . . . . . . . . . . . . . . . . . . 104

3.3.6 The case of divisors with normal crossings . . . . . . . . . 106

4 Holomorphic de Rham cohomology 109

4.1 Holomorphic de Rham cohomology . . . . . . . . . . . . . . . . . 109

4.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.1.2 Holomorphic differentials with log poles . . . . . . . . . . 111

4.1.3 GAGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.2 De Rham cohomology via the h′-topology . . . . . . . . . . . . . 113

4.2.1 h′-differentials . . . . . . . . . . . . . . . . . . . . . . . . 113

4.2.2 De Rham cohomology . . . . . . . . . . . . . . . . . . . . 114

4.2.3 GAGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5 The period isomorphism 117

5.1 The category (k,Q)−Vect . . . . . . . . . . . . . . . . . . . . . . 117

5.2 A triangulated category . . . . . . . . . . . . . . . . . . . . . . . 118

5.3 The period isomorphism in the smooth case . . . . . . . . . . . . 119

5.4 The general case (via the h′-topology) . . . . . . . . . . . . . . . 121

5.5 The general case (Deligne’s method) . . . . . . . . . . . . . . . . 123

II Nori Motives 127

6 Nori’s diagram category 129

6.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.1.1 Diagrams and representations . . . . . . . . . . . . . . . . 129

6.1.2 Explicit construction of the diagram category . . . . . . . 131

6.1.3 Universal property: Statement . . . . . . . . . . . . . . . 132

6.1.4 Discussion of the Tannakian case . . . . . . . . . . . . . . 135

6.2 First properties of the diagram category . . . . . . . . . . . . . . 136

6.3 The diagram category of an abelian category . . . . . . . . . . . 140

6.3.1 A calculus of tensors . . . . . . . . . . . . . . . . . . . . . 140

6 CONTENTS

6.3.2 Construction of the equivalence . . . . . . . . . . . . . . . 146

6.3.3 Examples and applications . . . . . . . . . . . . . . . . . 151

6.4 Universal property of the diagram category . . . . . . . . . . . . 153

6.5 The diagram category as a category of comodules . . . . . . . . . 156

6.5.1 Preliminary discussion . . . . . . . . . . . . . . . . . . . . 156

6.5.2 Coalgebras and comodules . . . . . . . . . . . . . . . . . . 157

7 More on diagrams 161

7.1 Multiplicative structure . . . . . . . . . . . . . . . . . . . . . . . 161

7.2 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

7.3 Nori’s Rigidity Criterion . . . . . . . . . . . . . . . . . . . . . . . 171

7.4 Comparing fibre functors . . . . . . . . . . . . . . . . . . . . . . 175

7.4.1 The space of comparison maps . . . . . . . . . . . . . . . 175

7.4.2 Some examples . . . . . . . . . . . . . . . . . . . . . . . . 180

7.4.3 The description as formal periods . . . . . . . . . . . . . . 183

8 Nori motives 187

8.1 Essentials of Nori Motives . . . . . . . . . . . . . . . . . . . . . . 187

8.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

8.1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . 188

8.2 Yoga of good pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 191

8.2.1 Good pairs and good filtrations . . . . . . . . . . . . . . . 191

8.2.2 Cech complexes . . . . . . . . . . . . . . . . . . . . . . . . 193

8.2.3 Putting things together . . . . . . . . . . . . . . . . . . . 195

8.2.4 Comparing diagram categories . . . . . . . . . . . . . . . 197

8.3 Tensor structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

8.3.1 Collection of proofs . . . . . . . . . . . . . . . . . . . . . . 203

III Periods 205

9 Periods of varieties 207

9.1 First definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

9.2 Periods for the category (k,Q)−Vect . . . . . . . . . . . . . . . . 210

9.3 Periods of algebraic varieties . . . . . . . . . . . . . . . . . . . . 213

CONTENTS 7

9.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

9.3.2 First properties . . . . . . . . . . . . . . . . . . . . . . . . 215

9.4 The comparison theorem . . . . . . . . . . . . . . . . . . . . . . . 216

10 Categories of mixed motives 219

10.1 Geometric motives . . . . . . . . . . . . . . . . . . . . . . . . . . 219

10.2 Absolute Hodge motives . . . . . . . . . . . . . . . . . . . . . . . 221

10.3 Comparison functors . . . . . . . . . . . . . . . . . . . . . . . . . 225

10.4 Weights and Nori motives . . . . . . . . . . . . . . . . . . . . . . 227

10.5 Periods of motives . . . . . . . . . . . . . . . . . . . . . . . . . . 229

11 Kontsevich-Zagier Periods 231

11.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

11.2 Comparison of Definitions of Periods . . . . . . . . . . . . . . . . 233

12 Formal periods and the period conjecture 241

12.1 Formal periods and Nori motives . . . . . . . . . . . . . . . . . . 241

12.2 The period conjecture . . . . . . . . . . . . . . . . . . . . . . . . 245

12.3 The case of 0-dimensional varieties . . . . . . . . . . . . . . . . . 249

IV Examples 253

13 Elementary examples 255

13.1 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

13.2 More Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

13.3 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

13.4 Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

13.5 Periods of 1-forms on arbitrary curves . . . . . . . . . . . . . . . 264

14 Multiple zeta values 269

14.1 A ζ-value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

14.2 Definition of multiple zeta values . . . . . . . . . . . . . . . . . . 271

14.3 Kontsevich’s integral representation . . . . . . . . . . . . . . . . 274

14.4 Shuffle and Stuffle relations for MZV . . . . . . . . . . . . . . . . 275

14.5 Multiple zeta values and moduli space of marked curves . . . . . 280

8 CONTENTS

14.6 Multiple Polylogarithms . . . . . . . . . . . . . . . . . . . . . . . 281

14.6.1 The Configuration . . . . . . . . . . . . . . . . . . . . . . 281

14.6.2 Singular Homology . . . . . . . . . . . . . . . . . . . . . . 282

14.6.3 Smooth Singular Homology . . . . . . . . . . . . . . . . . 285

14.6.4 Algebraic de Rham cohomology and period matrix of (X,D)286

14.6.5 Varying parameters a and b . . . . . . . . . . . . . . . . . 290

15 Miscellaneous periods: an outlook 295

15.1 Special values of L-functions . . . . . . . . . . . . . . . . . . . . . 295

15.2 Feynman periods . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

15.3 Algebraic cycles and periods . . . . . . . . . . . . . . . . . . . . . 300

15.4 Periods of homotopy groups . . . . . . . . . . . . . . . . . . . . . 302

15.5 Non-periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

V Bibliography 309

Part

Introduction

9

Chapter 0

Introduction

The aim of this book is to present the theory of period numbers and theirstructural properties. The second main theme is the theory of motives andcohomology which is behind these structural properties.

The genesis of this book is involved. Some time ago we were fascinated by astatement of Kontsevich [K1], stating that his algebra of formal periods is apro-algebraic torsor under the motivic Galois group of motives. He attributedthis theorem to Nori, but no proof was indicated.

We came to understand that it would indeed follow more or less directly fromNori’s unpublished description of an abelian category of motives. After realizingthis, we started to work out many details in our preprint [HMS] from 2011.

Over the years we have also realized that periods themselves generate a lotof interest, very often from non-specialists who are not familiar with all thetechniques going into the story. Hence we thought it would be worthwhile tomake this background accessible to a wider audience.

We started to write this monograph in a style suited also for non-expert readersby adding several introductory chapters and many examples.

0.1 General introduction

So what are periods?

A naive point of view

Period numbers are complex numbers defined as values of integrals∫γ

ω

11

12 CHAPTER 0. INTRODUCTION

of closed differential forms ω over certain domains of integration γ. One requiresrestrictive conditions on ω and γ, i.e., that γ is a region given by (semi)algebraicequations with rational coefficients, and the differential form ω is algebraic overQ. The analogous definition can be made for other fields, but we restrict to themain case k = Q in this introduction.

Many interesting numbers occuring in mathematics can be described in thisform.

1. log(2) is a period because2∫1

dxx = log(2).

2. π is a period because∫

x2+y2≤1

dxdy = π.

3. The Cauchy integral yields a complex period∫|z|=1

dz

z= 2πi .

4. Values of the Riemann zeta function like

ζ(3) =

∞∑n=1

1

n3=

∫0<x<y<z<1

dxdydz

(1− x)yz

are periods nubers as well.

5. More generally, all multiple zeta values (see Chapter 14) are period num-bers.

6. A basic observation is that all algebraic numbers are periods, e.g.,√

5 canobtained by integrating the differential form dx on the algebraic curvey = x2 over the real region where 0 ≤ y ≤ 5 and x ≥ 0.

Period numbers turn up in many parts of mathematics, sometimes in very sur-prising situations. Of course, they are a traditional object of number theoryand have been studied from different points of view. They also generate a lot ofinterest in mathematical physics because Feynman integrals for rational valuesof kinematical invariants are period numbers.

It is easy to write down periods. It is much harder to write down numbers whichare non-periods. This is surprising, given that the set of all period numbers isa countable algebra containing of Q. Indeed, we expect that π−1 and the Eulernumber e are non-periods, but this is not known. We refer to Section 15.5 foran actual, not too explicit example of a non-period.

It is as hard to understand linear or algebraic relations between periods. Thisaspect of the story starts with Lindemann’s 1882 proof of the transcendence ofπ and the transcendence of log(x) for x ∈ Q \ 0, 1. Grothendieck formulateda conjecture on the transcendence degree of the field generated by the periods

0.1. GENERAL INTRODUCTION 13

of any smooth projective variety. Historical traces of his ideas seem to go backat least to Leibniz, see Chapter 12. The latest development is Kontsevich’s for-mulation of a period conjecture for the algebra of all periods: the only relationsare the ones induced from the obvious ones, i.e., functoriality and long exactsequences in cohomology (see Chapter 12). The conjecture is very deep. As avery special case it implies the transcendence of ζ(n) for n odd. This is wideopen, the best available result being the irrationality of ζ(3)!

While this aspect is interesting and important, we really have almost nothingto say about it. Instead, we aim at explaining a more conceptual interpretationof period numbers and shed light on some structural properties of the algebraof periods numbers.

As an aside: Periods of integrals are also used in the theory of moduli of algebraicvarieties. Given a family of projective varieties, Griffiths defined a map into aperiod domain by studing the function given by varying period numbers. Weare not concerned with this point of view either.

A more conceptual point of view

The period integral∫γω actually only depends on the class of ω in de Rham

cohomology and on the class of γ in singular homology. Integration generalizesto the period pairing between algebraic de Rham cohomology and singular ho-mology. It has values in C, and the period numbers are precisely the image.Alternatively, one can formulate the relation as a period isomorphism betweenalgebraic de Rham cohomology and singular cohomology – after extension ofscalars to C. The comparison morphism is then described by a matrix whoseentries are periods. The most general situation one can allow here is relativecohomology of a possibly singular, possibly non-complete algebraic variety overQ with respect to a closed subvariety also defined over Q.

In formulas: For a variety X over Q, a closed subvariety Y over Q, and everyi ≥ 0, there is an isomorphism

per : HidR(X,Y )⊗Q C→ Hi

sing(Xan, Y an;Q)⊗Q C,

where Xan denotes the analytic space attached to X. If X is smooth, Xan issimply the complex manifold defined by the same equations as X. The reallyimportant thing to point out is the fact that this isomorphism does not respectthe Q-structures on both sides. Indeed, consider X = A1 \0 = SpecQ[T, T−1]and Y = ∅. The first de Rham cohomology group is one-dimensional andgenerated by dT

T . The first singular cohomology is also one-dimensional, andgenerated by the dual of the unit circle in Xan = C∗. The comparison factor isthe period integral

∫S1

dTT = 2πi.

Relative cohomology of pairs is a common standard in algebraic topology. Theanalogue on the de Rham side is much less so, in particular if X and Y arenot anymore smooth. Experts have been familiar with very general versions of


algebraic de Rham cohomology as by-products of advanced Hodge theory, butno elementary discussion seems to be in the literature. One of our intentions isto provide this here in some detail.

An even more conceptual point of view

An even better language to use is the language of motives. The concept wasintroduced by Grothendieck in his approach to Weil conjectures.

Motives are objects in a universal abelian category attached to the categoryof algebraic varieties whose most important property is to have cohomology:singular and de Rham cohomology in our case. Every variety has a motive h(X)which should decompose into components hi(X) for i = 1, . . . , 2 dimX. Singularcohomology of hi(X) is concentrated in degree i and equal to Hi

sing(Xan,Q)there.

Unfortunately, the picture still is largely conjectural. Pure motives – the onesattached to smooth projective varieties – have an unconditional definition dueto Grothendieck, but their expected properties depend on a choice of equiv-alence relations and hence on standard conjectures. In the mixed case – allvarieties – there are (at least) three candidates for an abelian category of mixedmotives (absolute Hodge motives of Deligne and Jannsen; Nori’s category; Ay-oub’s category). There are also a number of constructions of motivic triangu-lated categories (due to Hanamuara, Levine and Voevodsky) which we think ofas derived categories of the true category of mixed motives. They turn out tobe equivalent.

All standard properties of cohomology are assumed to be induced by propertiesof the category of motives: the Kunneth formula for the product of two vari-eties is induced by a tensor structure on motives; Poincare duality is inducedby the existence of strong duals on motives. In fact, every abelian categoryof motives (conjectural or candidate) is a rigid tensor category. Singular coho-mology is (supposed to be) a faithful and exact tensor functor on this tensorcategory. Hence, we have a Tannaka category. By the main theorem of Tan-naka theory, the category has a Tannaka dual: an affine pro-algebraic groupscheme whose finite dimensional representations are precisely mixed motives.This group scheme is the motivic Galois group Gmot.

This viewpoint allows a reinterpretation of the period algebra: singular and deRham cohomology are two fibre functors on the same Tannaka category, hencethere is a torsor of isomorphisms between them. The period isomorphism isnothing but a C-valued point of this torsor.

While the foundations of the theory of motives are still open, the good newsis that at least the definition of the period algebra does not depend on theparticular definition chosen. This is in fact one of the main results in thepresent book, see Chapter 10.5. Indeed, all variants of the definition yield thesame set of numbers, as we show in Part III. Among those are versions via

0.1. GENERAL INTRODUCTION 15

cohomology of arbitrary pairs of varieties, or only those of smooth varietiesrelative to divisors with normal crossings, or via semialgebraic simplices in Rn,and alternatively, with rational differential forms or only regular ones, and withrational or algebraic coefficients.

Nevertheless, the point of view of Nori’s category of motives turns out to beparticularly well-suited in order to treat periods. Indeed, the most naturalproof of the comparison results mentioned above is done in the language of Norimotives, see Chapter 12. This approach also fits nicely with the formulation ofthe period conjectures of Grothendieck and Kontsevich.

The period conjecture

Kontsevich in [K1] introduces a formal period algebra Peff where the Q-lineargenerators are given by quadruples (X,Y, ω, γ) with X an algebraic variety overQ, Y a closed subvariety, ω a class in Hn

dR(X,Y ) and γ ∈ Hsingn (Xan, Y n,Q).

There are three types of relations:

1. linearity in ω and γ;

2. functoriality with respect to morphisms f : (X,Y )→ (X ′, Y ), i.e.,

(X,Y, f∗ω, γ) ∼ (X ′, Y ′, ω, f∗γ);

3. compatibility with respect to connecting morphisms, i.e., for Z ⊂ Y ⊂ Xand δ : Hn−1

dR (Y,Z)→ HndR(X,Y )

(Y, Z, ω, ∂γ) ∼ (X,Y, δω, γ).

The set Peff becomes an algebra using the cup-product on cohomology. Therelations are defined in a way such that there is a natural evaluation map

Peff → C, (X,Y, ω, γ) 7→∫γ

ω.

(Actually this is a variant of the original definition, see Chapter 12.) In a secondstep, we localize with respect to the class of (A1 \ 0, 1, dT/T, S1), i.e., theformal period giving rise to 2πi. Basically by definition, the image of P is theperiod algebra.

Conjecture 0.1.1 (Period Conjecture, Kontsevich [K1]). The evaluation mapis injective.

Again, we have nothing to say about this conjecture. However, it shows thatthe elementary object P is quite natural in our context.

One of the main results in this book is the following result of Nori, which isstated already in [K1]


Theorem 0.1.2 (See Theorem 12.1.3). The formal period algebra P is a torsorunder the motivic Galois group in the sense of Nori, i.e., of the Tannaka dualof Nori’s category of motives.

Under the period conjecture, this should be read as a deep structural resultabout the period algebra.

Main aim of this book

We want to explain all the notions used above, give complete proofs, and discussa number of examples of particular interest.

• We explain singular cohomology and algebraic de Rham cohomology andthe period isomorphism between them.

• We introduce Nori’s abelian category of mixed motives and the necessarygeneralization of Tannaka theory going into the definition.

• Various notions of period numbers are introduced and compared.

• The relation of the formal period algebra to period numbers and the mo-tivic Galois group is explained.

• We work out examples like periods of curves, multiple zeta-values, Feyn-man integrals and special values of L-functions.

We strive for a reasonably self-contained presentation aimed also at non-specialistsand graduate students.

Relation to the existing literature

Both periods and the theory of motives have a long and rich history. We prefernot to attempt a historical survey, but rather mention the papers closest to thepresent book.

The definition of the period algebra was folklore for quite some time. Theexplicit versions we are treating are due to Kontsevich and Zagier in [K1] and[KZ].

Nori’s theory of motives became known through a series of talks that he gave,and notes of these talks that started to circulate, see [N], [N1]. Levine’s surveyarticle in [L1] sketches the main points.

The relation between (Nori) motives and formal periods is formulated by Kont-sevich [K1].

Finally, we would like to mention Andre’s monograph [A2]. Superficially, there isa lot of overlap (motives, Tannaka theory, periods). However, as our perspectiveis very different, we end up covering a lot of disjoint material as well.

0.2. RECENT DEVELOPMENTS 17

We recommend that anyone interested in a deeper understanding also study hisexposition.

0.2 Recent developments

The ideas of Nori have been taken up by other people in recent years, leadingto a rapid development of understanding. We have refrained from trying toincorporate all these results. It is too early to know what the final version ofthe theory will be. However, we would like to give at least some indication inwhich direction things are going.

The construction of Nori motives has been generalized to categories over a baseS by Arapura in [Ara] and Ivorra [Iv]. Arapura’s approach is based on con-structible sheaves. His categories allows pull-back and push-forward, the latterbeing a deep result. Ivorra’s approach is based on perverse sheaves. Compati-bility under the six functors formalism is open in his setting.

Harrer’s thesis [Ha] gives full proofs (based on the sketch of Nori in [N2]) of theconstruction of the realization functor from Voevodsky’s geometric motives toNori motives.

A comparison result of a different flavour was obtained by Choudhury and Gal-lauer [CG]: they are able to show that Nori’s motivic Galois group agrees withAyoub’s. The latter is defined via the Betti realization functor on triangulatedmotives over an arbitrary base. This yields formally a Hopf object in a derivedcategory of vector spaces. It is a deep result of Ayoub’s that the cohomology ofthis Hopf object is only concentrated in non-negative degrees. Hence its H0 isa Hopf algebra, the algebra of functions on Ayoub’s motivic Galois group.

The relation between these two objects, whose construction is very different,can be seen as a strong indication that Nori motives are really the true abeliancategory of mixed motives. One can strengthen this to the conjecture thatVoevodsky motives are the derived category of Nori motives.

In the same way as for other questions about motives, the case of 1-motives canbe hoped to be more accessible and a very good testing ground for this type ofcenjecture. Ayoub and Barbieri-Viale have shown in [AB] that the subcategoryof 1-motives in Nori motives agrees with Deligne’s 1-motives, and hence alsowith 1-motives in Voevodsky’s category.

There has also been progress on the period aspect of our book. Ayoub, in [Ay],proved a version of the period conjecture in families. There is also independentunpublished work of Nori on a similar question [N3].

We now turn to a more detailed description of the actual contents of our book.


0.3 Nori motives and Tannaka duality

Motives are supposed to be the universal abelian category over which all coho-mology theories factor. In this context, ”cohomology theory” means a (mixed)Weil cohomology theory with properties modeled on singular cohomology. Amore refined example of a mixed Weil cohomology theories is the mixed Hodgestructure on singular cohomology as defined by Deligne. Another one is `-adiccohomology of the base change of the variety to the algebraic closure of theground field. It carries a natural operation of the absolute Galois group of theground field. Key properties are for example a Kunneth formula for the productof algebraic varieties. There are other cohomology theories of algebraic varietieswhich do not follow the same pattern. Examples are algebraic K-theory, Delignecohomology or etale cohomology over the ground field. In all these cases theKunneth formula fails.

Coming back to theories similar to singular cohomology: they all take values inrigid tensor categories, and this is how the Kunneth formula makes sense. Weexpect the conjectural abelian category of mixed motives also to be a Tannakiancategory with singular cohomology as a fibre functor, i.e., a faithful exact tensorfunctor to Q-vector spaces. Nori takes this as the starting point of his definitionof his candidate for the category of mixed motives. His category is universal forall cohomology theories comparable to singular cohomology. This is not quitewhat we hope for, but it does in fact cover all examples we have.

Tannaka duality is built into the very definition. The construction has two mainsteps:

1. Nori first defines an abelian category which is universal for all cohomologytheories compatible with singular cohomology. By construction, it comeswith a functor on the category of pairs (X,Y ) where X is a variety andY a closed subvariety. Moreover, it is compatible with the long exactcohomology sequence for triples X ⊂ Y ⊂ Z.

2. He then introduces a tensor product and establishes rigidity.

The first step is completely formal and rests firmly on representation theory. Thesame construction can be made for any oriented graph and any representation ina category of modules over a noetherian ring. The abstract construction of this”diagram category” is explained in Chapter 6. Note that neither the tensorproduct nor rigidity is needed at this point. Nevertheless, Tannaka theoryis woven into proving that the diagram category has the necessary universalproperty: it is initial among all abelian categories over which the representationfactors. Looking closely at the arguments in Chapter 6, in particular Section6.3, we find the same arguments that are used in [DMOS] in order to establishthe existence of a Tannaka dual. In the case of a rigid tensor category, byTannaka duality it is equal to the category of representations of an affine groupscheme or equivalently co-representations of a Hopf algebra A. If we do not

0.3. NORI MOTIVES AND TANNAKA DUALITY 19

have rigidity, we do not have the antipodal map. We are left with a bialgebra.If we do not have a tensor product, we do not have a multiplication. We areleft with a coalgebra. Indeed, the diagram category can be described as theco-representations of an explicit coalgebra, if the coefficent ring is a Dedekindring or a field.

Chapter 7 aims at introducing a rigid tensor structure on the diagram category,or equivalently a Hopf algebra structure on the coalgebra. The product is in-duced by a product structure on the diagram and multiplicative representations.Rigidity is actually deduced as a property of the diagram category. Nori has astrong criterion for rigidity. Instead of asking for a unit and a counit, we onlyneed one of the two such that it becomes a duality under the representation.This rests on the fact that an algebraic submonoid of an algebraic group is analgebraic group. The argument is analogous to showing that a submonoid ofa finite abstract group is a group. Multiplication by an element is injective inthese cases, because it is injective on the group. If the monoid is finite, it alsohas to be surjective. Everything can also be applied to the diagram defined byany Tannaka category. Hence the exposition actually contains a full proof ofTannaka duality.

The second step is of completely different nature. It uses an insight on algebraicvarieties. This is the famous Basic Lemma of Nori, see Section 2.5. As it turnedout, Beilinson and also Vilonen had independently found the lemmma before.However, it was Nori who recognized its significance in such motivic situations.Let us explain the problem first. We would like to define the tensor productof two motives of the form Hn(X,Y ) and Hn′(X ′, Y ′). The only formula thatcomes to mind is

Hn(X,Y )⊗Hn′(X ′, Y ′) = HN (X ×X ′, X × Y ′ ∪ Y ×X ′)

with N = n + n′. This is, however, completely false in general. Cup-productwill give a map from the left to the right. By the Kunneth formula, we get anisomorphism when taking the sum over all n, n′ mit n+n′ = N on the left, butnot for a single summand.

Nori simply defines a pair (X,Y ) to be good, if its singular cohomology is con-centrated in a single degree and, moreover, a free module. In the case of goodpairs, the Kunneth formula is compatible with the naive tensor product of mo-tives. The Basic Lemma implies that the category of motives is generated bygood pairs. The details are explained in Chapter 8, in particular Section 8.2.

We would like to mention an issue that was particularly puzzling to us. Howis the graded commutativity of the Kunneth formula dealt with in Nori’s con-struction? This is one of the key problems in pure motives because it causessingular cohomology not to be compatible with the tensor structure on Chowmotives. The signs can be fixed, but only after assuming the Kunneth standardconjecture. Nori’s construction does not need to do anything about the prob-lem. So, how does it go away? The answer is the commutative diagram on p.165: the outer diagrams have signs, but luckily they cancel.


0.4 Cohomology theories

In Part I, we develop singular cohomology and algebraic de Rham cohomologyof algebraic varieties and the period isomorphism between them in some detail.

In Chapter 2, we recall as much of the properties of singular cohomology that isneeded in the sequel. We view it primarily as sheaf cohomology of the analyticspace associated to a variety over a fixed subfield k of C. In addition to standardproperties like Poincare duality and the Kunneth formula, we also discuss morespecial properties.

One such is Nori’s Basic Lemma: for a given affine variety X there is a closedsubvariety Y ⊂ such that relative cohomology is concentrated in a single degree.As discussed above, this is a crucial input for the construction of the tensorproduct on Nori motives. We give three proofs, two of them due to Nori, andan earlier one due to Beilinson.

In addition, in order to compare different possible definitions of the set of periodsnumbers, we need to understand triangulations of algebraic varieties by semi-algebraic simplices defined over Q.

Finally, we give a description of singular cohomology in terms of a Grothendiecktopology (the h′-topology) on analytic spaces which is used later in order todefine the period isomorphism.

Algebraic de Rham cohomology is much less documented in the literature.Through Hodge theory, the specialists have understood for a long time whatthe correct definition in the singular case are, but we are not aware of a co-herent exposition of algebraic de Rham cohomology by itself. This is whatChapter 3 is providing. We first treat systematically the more standard case ofa smooth variety where de Rham cohomology is given as hypercohomology ofthe de Rham complex. In a second step, starting in Section 3.2, we generalizeto the singular case. We choose the approach of the first author and Jorder in[HJ] via the h-cohomology on the category of k-varieties, but also explain therelation to Deligne’s approach via hypercovers and Hartshorne’s approach viaformal completion at the ideal of definition inside a smooth variety.

The final aim is to constract a natural isomorphism between singular cohomol-ogy and algebraic de Rham cohomology. This is established via the intermediatestep of holomorphic de Rham cohomology. The comparison between singularand holomorphic de Rham cohomology comes from the Poincare lemma: the deRham complex is a resolution of the constant sheaf. The comparison betweenalgebraic and holomorphic de Rham cohomology can be reduced to GAGA.This story is fairly well-known for smooth varieties. In our description with theh-topology, the singular case follows easily.

0.5. PERIODS 21

0.5 Periods

We have already discussed periods at some length at the beginning of the intro-duction. Roughly, a period number is the value of an integral of a differentialform over some algebraically defined domain. The definition can be made forany subfield k of C. There are several versions of the definition in the literatureand even more folklore versions around. They fall into three classes:

1. ”Naive” definitions have as domains of integration semi-algebraic simplicesin RN , over which one integrates rational differential forms defined over k(or over k), as long as the integral converges, see Chapter 11.

2. In more advanced versions, let X be an algebraic variety, and Y ⊂ X asubvariety, both defined over k, ω a closed algebraic differential form on Xdefined over k (or a de relative Rham cohomology class), and consider theperiod isomorphism between de Rham and singular cohomology. Periodsare the numbers coming up as entries of the period matrix. Variantsinclude the cases where X smooth, Y a divisor with normal crossings, orperhaps where X is affine, and smooth outside Y , see Chapter 9.

3. In the most sophisticated versions, take your favourite category of mixedmotives and consider the period isomorphism between their de Rham andsingular realization. Again, the entries of the period matrix are periods,see Chapter 10.

It is one of the main results of the present book that all these definitions agree. Adirect proof of the equivalence of the different versions of cohomological periodsis given in Chapter 9. A crucial ingredient of the proof is Nori’s descriptionof relative cohomology via the Basic Lemma. The comparison with periodsof geometric Voevodsky motives, absolute Hodge motives and Nori motives isdiscussed in Chapter 10. In Chapter 11, we discuss periods as in 1. and showthat they agree with cohomological periods.

The concluding Chapter 12 explains the deeper relation between periods of Norimotives and Kontsevich’s period conjecture, as already ementioned earlier in theintroduction. We also discuss the period conjecture itself.

0.6 Leitfaden

Part I, II, III and IV are supposed to be somewhat independent of each other,whereas the chapters in a each part depend more or less linearly on each other.

Part I is mostly meant as a reference for facts on cohomology that we need inthe development of the theory. Most readers will skip this part and only comeback to it when needed.


Part II is a self-contained introduction to the theory of Nori motives, where allparts build up on each other. Chapter 8 gives the actual definition. It needsthe input from Chapter 2 on singular cohomology.

Part III develops the theory of period numbers. Chapter 9 on cohomologicalperiods needs the period isomorphism of Chapter 5, and of course singularcohomology (Chapter 2) and algebraic de Rham cohomology (Chapter 3). Italso develops the linear algebra part of the theory of period numbers needed inthe rest of Part III. Chapter 10 has more of a survey character. It uses Norimotives, but should be understandable based just on the survey in Section 8.1.Chapter 11 is mostly self-contained, with some input from Chapter 9. Finally,Chapter 12 relies on the full force of the theory of Nori motives, in particularon the abstract results on the comparison of fibre functors in Section 7.4.

Part IV has a different flavour: Rather than developing theory, we go throughmany examples of period numbers. Actually, it may be a good starting pointfor reading the book or at least a good companion for the more general theorydeveloped in Part III.

I: Background Material

II: Nori Motives III: Period Numbers

IV: Examples

Ch. 2

Ch. 2,3,5

0.7 Acknowledgements

This work is fundamentally based on some unpublished work of M. Nori. Wethank him for several conversations. The presentation of his work in this bookis ours and hence, of course, all mistakes are ours.

Besides the preprint [HMS] of the main authors, this book is built on the work ofB. Friedrich [Fr] on periods and J. von Wangenheim [vW] on diagram categories.

0.7. ACKNOWLEDGEMENTS 23

We are very grateful to B. Friedrich and J. von Wangenheim for allowing us touse their work in this book. The material of Friedrich’s preprint is contained inSection 2.6, Chapters 9, 11, 13, and also 14. The diploma thesis of Wangenheimis basically Chapter 6.

Special thanks go to J. Ayoub, G. Wustholz for organizing with us the AlpbachWorkshop ”Motives, periods and transcendence” on [HMS] and related topicsin 2011. We thank all participants for their careful reading and subsequentcorrections. In particular, we would like to mention M. Gallauer, who found asevere flaw in Chapter 7 – and fixed it.

We have taken advantage from discussions and comments of Y. Andre, M. Gal-lauer, D. Harrer, F. Hormann, P. Jossen, S. Kebekus, D. van Straten, K. Volkel,and M. Wendt.

We are very grateful to W. Soergel for his continuous support in the Tannakianaspects of Part II. His insights greatly improved the exposition. We thankJ. Schurmann for his patient explanations on weakly constructible sheaves inSection 2.5. We would not have been able to find all these references withouthim.

We also thank all participants of the lectures on the topic held in Mainz and inFreiburg in 2014 and 2015.


Part I

Background Material

25

Chapter 1

General Set-up

In this chapter we collect some standard notation used throughout the book.

1.1 Varieties

Let k be field. It will almost always be of characteristic zero or even a subfieldof C.

By a scheme over k we mean a separated scheme of finite type over k. Let Schbe the category of k-schemes. By a variety over k we mean a quasi-projectivereduced scheme of finite type over k. Let Var be the category of varieties over k.Let Sm and Aff be the full subcategories of smooth varieties and affine varieties,respectively.

1.1.1 Linearizing the category of varieties

We also need the additive categories generated by these categories of varieties.More precisely:

Definition 1.1.1. Let Z[Var] be the category with objects the objects of Var.If X = X1∪· · ·∪Xn, Y = Y1∪· · ·∪Ym are varieties with connected componentsXi, Yj , we put

MorZ[Var](X,Y ) =

∑i,j

aijfij |aij ∈ Z, fij ∈ MorVar(Xi, Yj)

with the addition of formal linear combinations. Composition of morphisms isdefined by extending composition of morphisms of varieties Z-linearly.

Analogously, we define Z[Sm], Z[Aff] from Sm and Aff. Moreover, let Q[Var],

27

28 CHAPTER 1. GENERAL SET-UP

Q[Sm] and Q[Aff] be the analogous Q-linear additive categories where mor-phisms are formal Q-linear combinations of morphisms of varieties.

Let F =∑aifi : X → Y be a morphism in Z[Var]. The support of F is the set

of fi with ai 6= 0.

Z[Var] is an additive category with direct sum given by the disjoint union ofvarieties. The zero object corresponds to the empty variety, or, if you prefer,has to be added formally.

We are also going to need the category of smooth correspondences SmCor. Ithas the same objects as Sm and as morphisms finite correspondences

MorSmCor(X,Y ) = Cor(X,Y ),

where Cor(X,Y ) is the free Z-module with generators integral subschemes Γ ⊂X × Y such that Γ→ X is finite and dominant over a component of X.

1.1.2 Divisors with normal crossings

Definition 1.1.2. Let X be a smooth variety of dimension n and D ⊂ X aclosed subvariety. D is called divisor with normal crossings if for every pointof D there is an affine neighbourhood U of x in X which is etale over An viacoordinates t1, . . . , tn and such that D|U has the form

D|U = V (t1t2 · · · tr)

for some 1 ≤ r ≤ n.

D is called a simple divisor with normal crossings if in addition the irreduciblecomponents of D are smooth.

In other words, D looks etale locally like an intersection of coordinate hyper-planes.

Example 1.1.3. Let D ⊂ A2 be the nodal curve, given by the equation y2 =x2(x− 1). It is smooth in all points different from (0, 0) and looks etale locallylike xy = 0 in the origin. Hence it is a divisor with normal crossings but not asimple normal crossings divisor.

1.2 Complex analytic spaces

A classical reference for complex analytic spaces is the book of Grauert andRemmert [GR].

Definition 1.2.1. A complex analytic space is a locally ringed space (X,OholX )

with X paracompact and Hausdorff, and such that (X,OholX ) is locally isomor-

phic to the vanishing locus Z of a set S of holomorphic functions in some open

1.2. COMPLEX ANALYTIC SPACES 29

U ⊂ Cn and OholZ = Ohol

U /〈S〉, where OholU is the sheaf of holomorphic functions

on U .

A morphism of complex analytic spaces is a morphism f : (X,OholX )→ (Y,Ohol

Y )

of locally ringed spaces, which is given by a morphism of sheaves f : OholY →

f∗OholX that sends a germ h ∈ Ohol

Y,y of a holomorphic function h near y to thegerms h f , which are holomorphic near x for all x with f(x) = y. A morphismwill sometimes simply be called a holomorphic map, and will be denoted inshort form as f : X → Y .

Let An be the category of complex analytic spaces.

Example 1.2.2. Let X be a complex manifold. Then it can be viewed as acomplex analytic space. The structure sheaf is defined via the charts.

Definition 1.2.3. A morphism X → Y between complex analytic spaces iscalled proper if the preimage of any compact subset in Y is compact.

1.2.1 Analytification

Polynomials over C can be viewed as holomorphic functions. Hence an affinevariety immediately defines a complex analytic space. If X is smooth, it is evena complex submanifold. This assignment is well-behaved under gluing and henceit globalizes. A general reference for this is [SGA1], expose XII by M. Raynaud.

Proposition 1.2.4. There is a functor

·an : SchC → An

which assigns to a scheme of finite type over C its analytification. There is anatural morphism of locally ringed spaces

α : (Xan,OholXan)→ (X,OX)

and ·an is universal with this property. Moreover, α is the identity on points.

If X is smooth, then Xan is a complex manifold. If f : X → Y is proper, theso ist fan.

Proof. By the universal property it suffices to consider the affine case where theobvious construction works. Note that Xan is Hausdorff because X is separated,and it is paracompact because it has a finite cover by closed subsets of someCn. If X is smooth then Xan is smooth by [SGA1], Prop. 2.1 in expose XII,or simply by the Jacobi criterion. The fact that fan is proper if f is proper isshown in [SGA1], Prop. 3.2 in expose XII.


1.3 Complexes

1.3.1 Basic definitions

Let A be an additive category. If not specified otherwise, a complex will alwaysmean a cohomological complex, i.e., a sequence Ai for i ∈ Z of objects of Awith ascending differential di : Ai → Ai+1 such that di+1di = 0 for all i ∈ Z.The category of complexes is denoted by C(A). We denote C+(A), C−(A) andCb(A) the full subcategories of complexes bounded below, bounded above andbounded, respectively.

If K• ∈ C(A) is a complex, we define the shifted complex K•[1] with

(K•[1])i = Ki+1 , diK•[1] = −di+1K• .

If f : K• → L• is a morphism of complexes, its cone is the complex Cone(f)•

withCone(f)i = Ki+1 ⊕ Li, diCone(f) = (−di+1

K , f i+1 + diL) .

By construction there are morphisms

L• → Cone(f)→ K•[1] ,

Let K(A), K+(A), K−(A) and Kb(A) be the corresponding homotopy cate-gories where the objects are the same and morphisms are homotopy classes ofmorphisms of complexes. Note that these categories are always triangulatedwith the above shift functor and the class of distinguished triangles are thosehomotopy equivalent to

K•f−→ L• → Cone(f)→ K•[1]

for some morphism of complexes f .

Recall:

Definition 1.3.1. Let A be an abelian category. A morphism f• : K• → L•

of complexes in A is called quasi-isomorphism if

Hi(f) : Hi(K•)→ Hi(L•)

is an isomorphism for all i ∈ Z.

We will always assume that an abelian category has enough injectives, or isessentially small, in order to avoid set-theoretic problems. If A is abelian, letD(A), D+(A), D−(A) and Db(A) the induced derived categories where the ob-jects are the same as in K?(A) and morphisms are obtained by localizing K?(A)with respect to the class of quasi-isomorphisms. A triangle is distinguished if itis isomorphic in D?(A) to a distinguished triangle in K?(A).

Remark 1.3.2. Let A be abelian. If f : K• → L• is a morphism of complexes,then

0→ L• → Cone(f)→ K•[1]→ 0

is an exact sequence of complexes. Indeed, it is degreewise split-exact.

1.3. COMPLEXES 31

1.3.2 Filtrations

Filtrations on complexes are used in order to construct spectral sequences. Wemostly need two standard cases.

Definition 1.3.3. 1. Let A be an additive category, K• a complex in A.The stupid filtration F≥pK• on K• is given by

F≥pK• =

Ki i ≥ p,0 i < p.

The quotient K•/F≥pK• is given by

F<pK• =

0 i ≥ p,Ki i < p.

2. Let A be an abelian category, K• a complex in A. The canonical filtrationτ≤pK

• on K• is given by

F≤pK• =

Ki i < p,

Ker(dp) i = p,

0 i > p.

The quotient K•/F≤pK• is given by

τ>pK• =

0 i < p,

Kp/Ker(dp) i = p,

Ki i > p.

The associated graded pieces of the stupid filtration are given by

F≥pK•/F≥p+1K• = Kp .

The associated graded pieces of the canonical filtration are given by

τ≤pK•/τ≤p−1K

• = Hp(K•) .

1.3.3 Total complexes and signs

We return to the more general case of an additive category A. We considercomplexes in K•,• ∈ C(A), i.e., double complexes consisting of a set of objectsKi,j ∈ A for i, j ∈ Z with differentials

di,j1 : Ki,j → Ki,j+1 , di,j2 : Ki,j → Ki+1,j


such that (Ki,•, di,•2 ) and (K•,j , d•,j1 ) are complexes and the diagrams

Ki,j+1 di,j+12−−−−→ Ki+1,j+1

di,j1

x xdi+1,j1

Ki,j di,j2−−−−→ Ki+1,j

commute for all i, j ∈ Z. The associated simple complex or total complexTot(K•,•) is defined as

Tot(K•,•)n =⊕i+j=n

Ki,j , dnTot(K•,•) =∑i+j=n

(di,j1 + (−1)jdi,j2 ) .

In order to take the direct sum, either the category has to allow infinite directsums or we have to assume boundedness conditions to make sure that only finitedirect sums occur. This is the case if Ki,j = 0 unless i, j ≥ 0.

Examples 1.3.4. 1. Our definition of the cone is a special case: for f :K• → L•

Cone(f) = Tot(K•,•) , K•,−1 = K•, K•,0 = L• .

2. Another example is given by the tensor product. Given two complexes(F •, dF ) and (G•, dG), the tensor product

(F • ⊗G•)n =⊕i+j=n

F i ⊗Gj

has a natural structure of a double complex with Ki,j = F i⊗Gj , and thedifferential is given by d = idF ⊗ dG + (−1)idF ⊗ idF .

Remark 1.3.5. There is a choice of signs in the definition of the total complex.See for example [Hu1] §2.2 for a discussion. We use the convention opposite tothe one of loc. cit. For most formulae it does matter which choice is used,as long as it is used consistently. However, it does have an asymmetric effecton the formula for the compatibility of cup-products with boundary maps. Wespell out the source of this assymmetry.

Lemma 1.3.6. Let F •, G• be complexes in an additive tensor category. Then:

1. F • ⊗ (G•[1]) = (F • ⊗G•)[1].

2. ε : (F •[1] ⊗ G•) → (F • ⊗ G•)[1] with ε = (−1)j on F i ⊗ Gj (in degreei+ j − 1) is an isomorphism of complexes.

Proof. We compute the differential on F i⊗Gi in all three complexes. Note that

F i ⊗Gj = (F [1])i−1 ⊗Gj = F i ⊗ (G[1])j−1.

1.4. HYPERCOHOMOLOGY 33

For better readability, we drop ⊗id and id⊗ and |F i⊗Gj everywhere. Hence wehave

di+j−1(F•⊗G•)[1] = −di+jF•⊗G•

= −(djG• + (−1)jdiF•

)= −djG• + (−1)j−1diF•

di+j−1F•⊗(G•[1]) = dj−1

G•[1] + (−1)j−1diF•

= −djG• + (−1)j−1diF•

di+j−1(F•[1])⊗G• = djG• + (−1)jdi−1

F•[1]

= djG• + (−1)j−1diF•

We see that the first two complexes agree, whereas the differential of the thirdis different. Multiplication by (−1)j on the summand F i⊗Gj is a morphism ofcomplexes.

1.4 Hypercohomology

Let X be a topological space and Sh(X) the abelian category of sheaves ofabelian groups on X.

We want to extend the definition of sheaf cohomology on X, as explained in[Ha2], Chap. III, to complexes of sheaves.

1.4.1 Definition

Definition 1.4.1. Let F• be a bounded below complex of sheaves of abeliangroups on X. An injective resolution of F• is a quasi-isomorphism

F• → I•

where I• is a bounded below complex with In injective for all n, i.e., Hom(−, In)is exact.

Sheaf cohomology of X with coefficients in F• is defined as

Hi(X,F•) = Hi (Γ(X, I•)) i ∈ Z

where F• → I• is an injective resolution.

Remark 1.4.2. In the older literature, it is customary to write Hi(X,F•)instead of Hi(X,F•) and call it hypercohomology. We do not see any need todistinguish. Note that in the special case F• = F [0] a sheaf viewed as a complexconcentrated in degree 0, the notion of an injective resolution in the above senseagrees with the ordinary one in homological algebra.


Remark 1.4.3. In the language of derived categories, we have

Hi(X,F•) = HomD+(Sh(X))(Z,F•[i])

because Γ(X, ·) = HomSh(X)(Z, ·).

Lemma 1.4.4. Hi(X,F•) is well-defined and functorial in F•.

Proof. We first need existence of injective resolutions. Recall that the categorySh(X) has enough injectives. Hence every sheaf has an injective resolution. Thisextends to bounded below complexes in A by [We] Lemma 5.7.2 (or rather, itsanalogue for injective rather than projective objects).

Let F• → I• and G• → J • be injective resolutions. By loc.cit. Theorem 10.4.8

HomD+(Sh(X))(F•,G•) = HomK+(Sh(X))(I•,J •).

This means in particular that every morphism of complexes lifts to a morphismof injective resolutions and that the lift is unique up to homotopy of complexes.Hence the induced maps

Hi(Γ(X, I•))→ Hi(Γ(X,J •))

agree. This implies that Hi(X,F•) is well-defined and a functor.

Remark 1.4.5. Injective sheaves are abundant (by our general assumptionthat there are enough injectives), but not suitable for computations. Everyinjective sheaf F is flasque [Ha1, III. Lemma 2.4], i.e., the restriction mapsF(U) → F(V ) between non-empty open sets V ⊂ U are always surjective.Below we will introduce the canonical flasque Godemont resolution for any sheafF . More generally, every flasque sheaf F is acyclic, i.e., Hi(X,F) = 0 for i > 0.One may compute sheaf cohomology of F using any acyclic resolution F •. Thisfollows from the hypercohomology spectral sequence

Ep,q2 = Hp(Hq(F •))⇒ Hp+q(X,F)

which is supported entirely on the q = 0-line.

Special acylic resolutions on X are the so-called fine resolutions. See [Wa,pg. 170] for a definition of fine sheaves involving partitions of unity. Theirimportance comes from the fact that sheaves of C∞-functions and sheaves ofC∞-differential forms on X are fine sheaves.

1.4.2 Godement resolutions

For many purposes, it is useful to have functorial resolutions of sheaves. Onesuch is given by the Godement resolution introduced in [God] chapter II §3.

Let X be a topological space. Recall that a sheaf on X vanishes if and only thestalks at all x ∈ X vanish. For all x ∈ X we denote ix : x → X the naturalinclusion.


Definition 1.4.6. Let F ∈ Sh(X). Put

I(F) =∏x∈X

ix∗Fx .

Inductively, we define the Godement resolution Gd•(F) of F by

Gd0(F) = I(F) ,

Gd1(F) = I(Coker(F → Gd0(F))) ,

Gdn+1(F) = I(Coker(Gdn−1(F)→ Gdn(F))) n > 0.

Lemma 1.4.7. 1. Gd is an exact functor with values in C+(Sh(X)).

2. The natural map F → Gd•(F) is a flasque resolution.

Proof. Functoriality is obvious. The sheaf I(F) is given by

U 7→∏x∈U

ix∗Fx .

All the sheaves involved are flasque, hence acyclic. In particular, taking thedirect products is exact (it is not in general), turning I(F) into an exact functor.F → I(F) is injective, and hence by construction, Gd•(F) is then a flasqueresolution.

Definition 1.4.8. Let F• ∈ C+(Sh(X)) be a complex of sheaves. We call

Gd(F•) := Tot(Gd•(F•))

the Godement resolution of F•.

Corollary 1.4.9. The natural map

F → Gd(F•)

is a quasi-isomorphism and

Hi(X,F•) = Hi (Γ(X,Gd(F•))) .

Proof. By Lemma 1.4.7, the first assertion holds if F• is concentrated in a singledegree. The general case follows by the hypercohomology spectral sequence orby induction on the length of the complex using the stupid filtration.

All terms in Gd(F•) are flasque, hence acyclic for Γ(X, ·).

We now study functoriality of the Godement resolution. For a continuous mapf : X → Y be denote f−1 the pull-back functor on sheaves of abelian groups.Recall that it is exact.


Lemma 1.4.10. Let f : X → Y be a continuous map between topological spaces,F• ∈ C+(Sh(Y )). Then there is a natural quasi-isomorphism

f−1GdY (F•)→ GdX(f−1F•) .

Proof. Consider a sheaf F on Y . We want to construct

f−1I(F)→ I(f−1F) =∏x∈X

ix∗(f−1F)x =

∏x∈X

ix∗Ff(x) .

By the universal property of the direct product and adjunction for f−1, this isequivalent to specifying for all x ∈ X∏

y∈Yiy∗Fy = I(F)→ f∗ix∗Ff(x) = if(x)∗Ff(x) .

We use the natural projection map. By construction, we have a natural com-mutative diagram

f−1F −−−−→ f−1I(F) −−−−→ Coker(f−1F → f−1I(F)

)=

y yf−1F −−−−→ I(f−1F) −−−−→ Coker

(f−1F → I(f−1F)

)It induces a map between the cokernels. Proceeding inductively, we obtain amorphism of complexes

f−1Gd•Y (F)→ Gd•X(f−1F) .

It is a quasi-isomorphism because both are resolutions of f−1F . This transfor-mation of functors extends to double complexes and hence defines a transfor-mation of functors on C+(Sh(Y )).

Remark 1.4.11. We are going to apply the theory of Godement resolutions inthe case where X is a variety over a field k, a complex manifold or more generallya complex analytic space. The continuous maps that we need to consider arethose in these categories, but also the maps of schemes XK → Xk for the changeof base field K/k and a variety over k; and the continuous map Xan → X foran algebraic variety over C and its analytification.

1.4.3 Cech cohomology

Neither the definition of sheaf cohomology via injective resolutions nor Gode-ment resolutions are convenient for concrete computations. We introduce Cechcohomology for this task. We follow [Ha2], Chap. III §4, but extend to hyper-cohomology.


Let k be a field. We work in the category of varieties over k. Let I = 1, . . . , nas ordered set and U = Ui|i ∈ I an affine open cover of X. For any subsetJ ⊂ 1, . . . , n we denote

UJ =⋂j∈J

Uj .

As X is separated, they are all affine.

Definition 1.4.12. Let X and U be as above. Let F ∈ Sh(X). We define theCech complex of F as

Cp(U,F) =∏

J⊂I,|J|=p+1

F(UJ) p ≥ 0

with differential δp : Cp(U,F)→ Cp+1(U,F)

(δpα)i0<i1<···<ip =

p+1∑j=0

(−1)pαi0···<ij<···<ip+1|Ui0...ip+1

,

where, as usual, i0 · · · < ij < · · · < ip+1 means the tuple with ij removed.

We define the p-th Cech cohomology of X with coefficients in F as

Hp(U,F) = Hp(C•(U,F), δ) .

Proposition 1.4.13 ([Ha2], chap. III Theorem 4.5). Let X be a variety, U anaffine open cover as before. Let F be a coherent sheaf of OX-modules on X.Then there is a natural isomorphism

Hp(X,F) = Hp(U,F) .

We now extend to complexes. We can apply the functor C•(U, ·) to all terms ina complex F• and obtain a double complex C•(U,F•).

Definition 1.4.14. Let X and U as before. Let F• ∈ C+(Sh(X)). We definethe Cech complex of U with coefficients in F• as

C•(U,F•) = Tot (C•(U,F•))

and Cech cohomology as

Hp(U,F) = Hp(C•(U,F•)) .

Proposition 1.4.15. Let X be a variety, U as before an open affine cover ofX. Let F• ∈ C+(Sh(X)) be complex such that all Fn are coherent sheaves ofOX-modules. Then there is a natural isomorphism

Hp(X,F)→ Hp(U,F•) .


Proof. The essential step is to define the map. We first consider a single sheafG. Let C•(U,G) be a sheafified version of the Cech complex for a sheaf G. By[Ha2], chap. III Lemma 4.2, it is a resolution of G. We apply the Godementresolution and obtain a flasque resolution of G by

G → C•(U,G)→ Gd (C•(U,G)) .

By Proposition 1.4.13, the induced map

C•(U,G)→ Γ(X,Gd (C•(U,G))

is a quasi-isomorphism as both compute Hi(X,G).

The construction is functorial in G, hence we can apply it to all components ofa complex F• and obtain double complexes. We use the previous results for allFn and take total complexes. Hence

F• → TotC•(U,F•)→ Gd (C•(U,F•))

are quasi-isomorphisms. Taking global sections we get a quasi-isomorphism

TotC•(U,F•)→ TotΓ(X,Gd (C•(U,F•))) .

By definition, the complex on the left computes Cech cohomology of F• andthe complex on right computes hypercohomology of F•.

Corollary 1.4.16. Let X be an affine variety and F• ∈ C+(Sh(X)) such thatall Fn are coherent sheaves of OX-modules. Then

Hi(Γ(X,F•)) = Hi(X,F•) .

Proof. We use the affine covering U = X and apply Proposition 1.4.15.

1.5 Simplicial objects

We introduce simplicial varieties in order to carry out some of the constructionsin [D5]. Good general references on the topic are [May] or [We] Ch. 8.

Definition 1.5.1. Let ∆ be the category whose objects are finite ordered sets

[n] = 0, 1, . . . , n n ∈ N0

with morphisms nondecreasing monotone maps. Let ∆N be the full subcategorywith objects the [n] with n ≤ N .

If C is a category, we denote by C∆ the category of simplicial objects in C definedas contravariant functors

X• : ∆→ C

1.5. SIMPLICIAL OBJECTS 39

with transformation of functors as morphisms. We denote by C∆ the categoryof cosimplicial objects in C defined as covariant functors

X• : ∆→ C .

Similarly, we defined the categories C∆N and C∆N of N -truncated simplicial andcosimplicial objects.

Example 1.5.2. Let X be an object of C. The constant functor

∆ → C

which maps all objects to X and all morphism to the identity morphism is asimplicial object. It is called the constant simplicial object associated to X.

In ∆, we have in particular the face maps

εi : [n− 1]→ [n] i = 0, . . . , n,

the unique injective map leaving out the index i, and the degeneracy maps

ηi : [n+ 1]→ [n] i = 0, . . . , n,

the unique surjective map with two elements mapping to i. More generally, amap in ∆ is called face or degeneracy if it is a composition of εi or ηi, respectively.Any morphism in ∆ can be decomposed into a degeneracy followed by a face([We] Lemma 8.12).

For all m ≥ n, we denote Sm,n the set of all degeneracy maps [m]→ [n].

A simplicial object X• is determined by a sequence of objects

X0, X1, . . .

and face and degeneracy morphisms between them. In particular, we write

∂i : Xn → Xn−1

for the image of εi andsi : Xn → Xn+1

for the image of ηi.

Example 1.5.3. For every n, there is a simplicial set ∆[n] with

∆[n]m = Mor∆([m], [n])

and the natural face and degeneracy maps. It is called the simplicial n-simplex.For n = 0, this is the simplicial point, and for n = 1 the simplicial interval.Functoriality in the first argument induces maps of simplicial sets. In particular,there are

δ0 = ε∗0, δ1 = ε∗1 : ∆[1]→ ∆[0] .


Definition 1.5.4. Let C be a category with finite products and coproducts.Let ? be the final object. Let X•, Y• simplicial objects in C and S• a simplicialset

1. X• × Y• is the simplicial object with

(X• × Y•)n = Xn × Yn

with face and degeneracy maps induced from X• and Y•.

2. X• × S• is the simplicial object with

(X• × S•)n =∐s∈Sn

Xn

with face and degeneracy maps induced from X• and S•.

3. Let f, g : X• → Y• be morphisms of simplicial objects. Then f is calledhomotopic to g if there is a morphism

h : X• ×∆[1]→ Y•

such that h δ0 = f and h δ1 = g.

The inclusion ∆N → ∆ induces a natural restriction functor

sqN : C∆ → C∆N .

For a simplicial object X•, we call sqNX• its N -skeleton. If Y• is a fixed simpli-cial objects, we also denote sqN the restriction functor from simplicial objectsover Y• to simplicial objects over sqNY•.

Remark 1.5.5. The skeleta sqkX• define the skeleton filtration, i.e., a chain ofmaps

sq0X• → sq1X• → · · · → sqNX• = X•.

Later, in section 2.3, we will define the topological realization |X•| of a simpli-cial set X•. The skeleton filtration then defines a filtration of |X•| by closedsubspaces.

An important example in this book is the case when the simplicial set X• is afinite set, i.e., all Xn are finite sets, and empty for n > N sufficiently large. Seesection 2.3.

Lemma 1.5.6. Let C be a category with finite limits. Then the functor sqN hasa right adjoint

cosqN : C∆N → C∆ .

If Y• is a fixed simplicial object, then

cosqY•N (X•) = cosqNX× ×cosqN sqNY• Y•

is the right adjoint of the relative version of sqN .

1.5. SIMPLICIAL OBJECTS 41

Proof. The existence of cosqN is an instance of a Kan extension. We refer to[ML, chap. X] or [AM, chap. 2] for its existence. The relative case follows fromthe universal properties of fibre products.

If X• is an N -truncated simplicial object, we call cosqNX• its coskeleton.

Remark 1.5.7. We apply this in particular to the case where C is one of thecategories Var, Sm or Aff over a fixed field k. The disjoint union of varieties isa coproduct in these categories and the direct product a product.

Definition 1.5.8. Let S be a class of covering maps of varieties containingall identity morphisms. A morphism f : X• → Y• of simplicial varieties (orthe simplicial variety X• itself) is called an S-hypercovering if the adjunctionmorphisms

Xn → (cosqY•n−1sqn−1X•)n

are in S.

If S is the class of proper, surjective morphisms, we call X• a proper hypercoverof Y•.

Definition 1.5.9. Let X• be a simplical variety. It is called split if for alln ∈ N0

N(Xn) = Xn rn−1⋃i=0

si(Xn−1)

is an open and closed subvariety of Xn.

We call N(Xn) the non-degenerate part of Xn. If X• is a split simplicial variety,we have a decomposition as varieties

Xn = N(Xn)q∐m<n

∐s∈Sm,n

sN(Xm)

where Sm,n is the set of degeneracy maps from Xm to Xn.

Theorem 1.5.10 (Deligne). Let k be a field and Y a variety over k. Thenthere is a split simplicial variety X• with all Xn smooth and a proper hypercoverX• → Y .

Proof. The construction is given in [D5] Section (6.2.5). It depends on theexistence of resolutions of singularities. In positive characteristic, we may usede Jong’s result on alterations instead.

The other case we are going to need is the case of additive categories.

Definition 1.5.11. Let A be an additive category. We define a functor

C : A∆ → C−(A)


by mapping a simplicial object X• to the cohomological complex

. . . X−nd−n−−→ X−(n−1) → · · · → X0 → 0

with differential

d−n =

n∑i=0

(−1)i∂i .

We define a functorC : A∆ → C+(A)

by mapping a cosimplicial object X• to the cohomological complex

0→ X0 → . . . Xn dn−→ Xn+1 → . . .

with differential

dn =

n∑i=0

(−1)i∂i .

Let A be an abelian category. We define a functor

N : A∆ → C+(A)

by mapping a cosimplicial object X• to the normalized complex N(X•) with

N(X•)n =

n−1⋂i=0

Ker(si : Xn → Xn−1)

and differential dn|N(X•).

Proposition 1.5.12 (Dold-Kan correspondence). Let A be an abelian category,X• ∈ A∆ a cosimplicial object. Then the natural map

N(X•)→ C(X•)

is a quasi-isomorphism.

Proof. This is the dual result of [We], Theorem 8.3.8.

Remark 1.5.13. We are going to apply this in the case of cosimplicial com-plexes, i.e., to C(A)∆ , where A is abelian, e.g., a category of vector spaces.

1.6 Grothendieck topologies

Grothendieck topologies generalize the notion of open subsets in topologicalspaces. Using them one can define cohomology theories in more abstract set-tings. To define a Grothendieck topology, we need the notion of a site, or situs.Let C be a category. A basis for a Grothendieck topology on C is given bycovering families in the category C satisfying the following definition.

1.6. GROTHENDIECK TOPOLOGIES 43

Definition 1.6.1. A site/situs is a category C together with a collection ofmorphism in C

(ϕi : Vi −→ U)i∈I ,

the covering families.

The covering families satisfy the following axioms:

• An isomorphism ϕ : V → U is a covering family with an index set con-taining only one element.

• If (ϕi : Vi −→ U)i∈I is a covering family, and f : V → U a morphism inC, then for each i ∈ I there exists the pullback diagram

V ×U ViFi−−−−→ Vi

Φi

y yϕiV

f−−−−→ U

in C, and (Φi : V ×U Vi → V )i∈I is a covering family of V .

• If (ϕi : Vi −→ U)i∈I is a covering family of U , and for each Vi there is

given a covering family(ϕij : V ij → Vi

)j∈J(i)

, then

(ϕi ϕij : V ij → U

)i∈I,j∈J(i)

is a covering family of U .

Example 1.6.2. Let X be a topological space. Then the category of opensets in X together with inclusions as morphisms form a site, where the coveringmaps are the families (Ui)i∈I of open subsets of U such that ∪i∈IUi = U . Thuseach topological space defines a canonical site. For the Zariski open subsets ofa scheme X this is called the (small) Zariski site of X.

Definition 1.6.3. A presheaf F of abelian groups on a situs C is a contravariantfunctor

F : C → Ab, U 7→ F(U).

A presheaf F is a sheaf, if for each covering family (ϕi : Vi −→ U)i∈I , the dif-ference kernel sequence

0→ F(U)→∏i∈IF(Vi) ⇒

∏(i,j)∈I×I

F(Vi ×U Vj)

is exact. This means that a section s ∈ F(U) is determined by its restrictionsto each Vi, and a tuple (si)i∈I of sections comes from a section on U , if one hassi = sj on pullbacks to the fiber product Vi ×U Vj .


Once we have a notion of sheaves in a certain Grothendieck topology, then wemay define cohomology groups H∗(X,F) by using injective resolutions as insection 1.4 as the right derived functor of the left-exact global section functorX 7→ F(X) = H0(X,F) in the presence of enough injectives.

Example 1.6.4. The (small) etale site over a smooth variety X consists of thecategory of all etale morphisms ϕ : U → X from a smooth variety U to X. See[Ha2, Chap. III] for the notion of etale maps. We just note here that etale mapsare quasi-finite, i.e., have finite fibers, are unramified, and the image ϕ(U) ⊂ Xis a Zariski open subset.

A morphism in this site is given by a commutative diagram

Vf−−−−→ Uy y

Xid−−−−→ X.

Let U be etale over X. A family (ϕi : Vi −→ U)i∈I of etale maps over X iscalled a covering family of U , if

⋃i∈I ϕi(Vi) = U , i.e., the images form a Zariski

open covering of U .

This category has enough injectives by Grothendieck [SGA4.2], and thus we candefine etale cohomology H∗et(X,F) for etale sheaves F .

Example 1.6.5. In Section 2.7 we are going to introduce the h′-topology onthe category of analytic spaces.

Definition 1.6.6. Let C and C′ be sites. A morphism of sites f : C → C′consists of a functor F : C′ → C (sic) which preserves fibre products and suchthat the F applied to a covering family of C′ yields a covering family of C.

A morphism of sites induces an adjoint pair of functors (f∗, f∗) between sheavesof sets on C and C′.

Example 1.6.7. 1. Let f : X → Y be continuous map of topological spaces.As in Example 1.6.2 we view them as sites. Then the functor F mappingan open subset of Y to its preimage f−1(U).

2. Let X be a scheme. Then there is morphism of sites from the small etalesite of X to the Zariki-site of X. The functor views an open subschemeU ⊂ X as an etale X-scheme. Open covers are in particular etale covers.

Definition 1.6.8. Let C be a site. A C-hypercover is an S-hypercover in thesense of Definition 1.5.8 with S the class of morphism∐

i∈IUi → U

for all covering families φi : Ui → Ui∈I in the site.

1.7. TORSORS 45

If X• is a simplical object and F is a presheaf of abelian groups, then F(X•)is a cosimplicial abelian group. By applying the total complex functor C ofDefinition 1.5.11, we get a complex of abelian groups.

Proposition 1.6.9. Let C be a site closed under finite products and fibre prod-ucts, F a sheaf of abelian groups on C, X ∈ C. Then

Hi(X,F) = limX•→X

Hi (C(F(X•)))

where the direct limit runs through the system of all C-hypercovers of X.

Proof. This is [SGA4V, Theoreme 7.4.1]

1.7 Torsors

Informally, a torsor is a group without a unit. The standard notion in algebraicgeometry is sheaf theoretic: a torsor under a sheaf of groups G is a sheaf ofsets X with an operation G × X → X such that there is a cover over whichX becomes isomorphic to G and the action becomes the group operation. Thismakes sense in any site.

In this section, we are going to discuss a variant of this idea which does notinvolve sites or topologies but rather schemes. This approach fits well with theTannaka formalism that will be discussed in Chapters 7.4 and 12.

It is used by Kontsevich in [K1]. This notion at least goes back to a paper ofR. Baer [Ba] from 1929, see the footnote on page 202 of loc. cit. where Baerexplains how the notion of a torsor comes up in the context of earlier work of H.Prufer [Pr]. In yet another context, ternary operations satisfying these axiomsare called associative Malcev operations, see [Joh] for a short account.

1.7.1 Sheaf theoretic definition

Definition 1.7.1. Let C be a category equipped with a Grothendieck topologyt. Assume S is a final object of C. Let G be a group object in C. A (left)G-torsor is an object X with a (left) operation

µ : G×X → X

such that there is a t-covering U → S such that restriction of G and X to U isthe trivial torsor, i.e., X(U) is non-empty, and the choice x ∈ X(U) induces anatural isomorphism

·x : G(U ′)→ X(U ′)

g 7→ µ(g, x).

for all U ′ → U .


The condition can also be formulated as an isomorphism

G× U → X × U(g, u) 7→ g(u), u)

Remark 1.7.2. 1. As µ is an operation, the isomorphism of the definitionis compatible with the operation as well, i.e., the diagram

G(U ′)×X(U ′)µ // X(U ′)

G(U ′)×G(U ′) //

(id,·x)

OO

G(U ′)

·x

OO

commutes.

2. If, moreover, X → S is a t-cover, then X(X) is always non-empty andwe recover a version of the definition that often appears in the literature,namely that

G×X → X ×Xhas to be an isomorphism.

We are interested in the topology that is in use in Tannaka theory. It is basicallythe flat topology, but we have to be careful what we mean by this because theschemes involved are not of finite type over the base.

Definition 1.7.3. Let S be an affine scheme and C the category of affine S-schemes. The fpqc-topology on C is generated by covers of the form X → Ywith O(X) faithfully flat over O(Y ).

The letters fpqc stand for fidelement plat quasi-compact. Recall that SpecA isquasi-compact for all rings A.

We do not discuss the non-affine case at all, but see the survey [Vis] by Vistolifor the general case. The topology is discussed in loc. cit. Section 2.3.2. Theabove formulation follows from loc. cit. Lemma 2.60.

Remark 1.7.4. If, moreover, S = Spec(k) is the spectrum of a field, then anynon-trivial SpecA→ Spec(k) is an fpqc-cover. Hence, we are in the situation ofRemark 1.7.2. Note that X still has to be non-empty!

The importance of the fpqc-topology is that all representable presheaves arefpqc-sheaves, see [Vis, Theorem 2.55].

1.7.2 Torsors in the category of sets

Definition 1.7.5 ([Ba] p. 202, [K1] p. 61, [Fr] Definition 7.2.1). A torsor is aset X together with a map

(·, ·, ·) : X ×X ×X → X

1.7. TORSORS 47

satisfying:

1. (x, y, y) = (y, y, x) = x for all x, y ∈ X

2. ((x, y, z), u, v) = (x, (u, z, y), v) = (x, y, (z, u, v)) for all x, y, z, u, v ∈ X.

Morphisms are defined in the obvious way, i.e., maps X → X ′ of sets commutingwith the torsor structure.

Lemma 1.7.6. Let G be a group. Then (g, h, k) = gh−1k defines a torsorstructure on G.

Proof. This is a direct computation:

(x, y, y) = xy−1y = x = yy−1x = (y, y, x),

((x, y, z), u, v) = (xy−1z, u, v) = xy−1zu−1v = (x, y, zu−1v) = (x, y, (z, u, v)),

(x, (u, z, y), v) = (x, uz−1y, v) = x(uz−1y)−1v) = xy−1zu−1v.

Lemma 1.7.7 ([Ba] page 202). Let X be a torsor, e ∈ X an element. ThenGe := X carries a group structure via

gh := (g, e, h), g−1 := (e, g, e).

Moreover, the torsor structure on X is given by the formula (g, h, k) = gh−1kin Ge.

Proof. First we show associativity:

(gh)k = (g, e, h)k = ((g, e, h), e, k) = (g, e, (h, e, k)) = g(h, e, k) = g(hk).

e becomes the neutral element:

eg = (e, e, g) = g; ge = (g, e, e) = g.

We also have to show that g−1 is indeed the inverse element:

gg−1 = g(e, g, e) = (g, e, (e, g, e)) = ((g, e, e), g, e) = (g, g, e) = e.

Similarly one shows that g−1g = e. One gets the torsor structure back, since

gh−1k = g(e, h, e)k = (g, e, (e, h, e))k = ((g, e, (e, h, e)), e, k)

= (g, (e, (e, h, e), e), k) = (g, ((e, e, h), e, e), k)

= (g, (h, e, e), k) = (g, h, k).


Proposition 1.7.8. Let µl : X2 ×X2 → X2 be given by

µl ((a, b), (c, d)) = ((a, b, c), d).

Then µl is associative and has (x, x) for x ∈ X as left-neutral elements. LetGl = X2/ ∼l where (a, b) ∼l (a, b)(x, x) for all x ∈ X is an equivalence relation.Then µl is well-defined on Gl and turns Gl into a group. Moreover, the torsorstructure map factors via a simply transitive left Gl-operation on X which isdefined by

(a, b)x := (a, b, x).

Let e ∈ X. Then

ie : Ge → Gl, x 7→ (x, e)

is group isomorphism inverse to (a, b) 7→ (a, b, e).In a similar way, using µr ((a, b), (c, d)) := (a, (b, c, d)) we obtain a group Gr

with analogous properties acting transitively on the right on X and such that µrfactors through the action X ×Gr → X.

Proof. First we check associativity of µl:

(a, b)[(c, d)(e, f)] = (a, b)((c, d, e), f) = ((a, b, (c, d, e)), f) = (((a, b, c), d, e), f)

[(a, b)(c, d)](e, f) = ((a, b, c), d)(e, f) = (((a, b, c), d, e), f)

(x, x) is a left neutral element for every x ∈ X:

(x, x)(a, b) = ((x, x, a), b) = (a, b)

We also need to check that ∼l is an equivalence relation: ∼l is reflexive, since onehas (a, b) = ((a, b, b), b) = (a, b)(b, b) by the first torsor axiom and the definitionof µ. For symmetry, assume (c, d) = (a, b)(x, x). Then

(a, b) = ((a, b, b), b) = ((a, b, (x, x, b)), b) = (((a, b, x), x, b), b)

= ((a, b, x), x)(b, b) = (a, b)(x, x)(b, b) = (c, d)(b, b)

again by the torsor axioms and the definition of µl. For transitivity observethat

(a, b)(x, x)(y, y) = (a, b)((x, x, y), y) = (a, b)(y, y).

Now we show that µl is well-defined on Gl:

[(a, b)(x, x)][(c, d)(y, y)] = (a, b)[(x, x)(c, d)](y, y) = (a, b)(c, d)(y, y).

The inverse element to (a, b) in Gl is given by (b, a), since

(a, b)(b, a) = ((a, b, b), a) = (a, a).

1.7. TORSORS 49

Define the left Gl-operation on X by (a, b)x := (a, b, x). This is compatible withµl, since

[(a, b)(c, d)]x = ((a, b, c), d)x = ((a, b, c), d, x),

(a, b)[(c, d)x] = (a, b)(c, d, x) = ((a, b, (c, d, x))

are equal by the second torsor axiom. The left Gl-operation is well-defined withrespect to ∼l:

[(a, b)(x, x)]y = ((a, b, x), x)y = ((a, b, x), x, y) = (a, (x, x, b), y) = (a, b, y) = (a, b)y.

Now we show that ie is a group homomorphism:

ab = (a, e, b) 7→ ((a, e, b), e) = (a, e)(b, e)

The inverse group homomorphism is given by

(a, b)(c, d) = ((a, b, c), d) 7→ ((a, b, c), d, e).

On the other hand in Ge one has:

(a, b, e)(c, d, e) = ((a, b, e), e, (c, d, e)) = (a, b, (e, e, (c, d, e))) = (a, b, (c, d, e)).

This shows that ie is an isomorphism. The fact that Ge is a group implies thatthe operation of Gl on X is simply transitive. Indeed the group structure onGe = X is the one induced by the operation of Gl. The analogous group Gr isconstructed using µr and an equivalence relation ∼r with opposite order, i.e.,(a, b) ∼r (x, x)(a, b) for all x ∈ X. The properties of Gr can be verified in thesame way as for Gl and are left to the reader.

1.7.3 Torsors in the category of schemes (without groups)

Definition 1.7.9. Let S be a scheme. A torsor in the category of S-schemesis a non-empty scheme X and a morphism

X ×X ×X → X

which on T -valued points is a torsor in the sense of Definition 1.7.5 for all Tover S.

This simply means that the diagrams of the previous definition commute as mor-phisms of schemes. The following is the scheme theoretic version of Lemma 1.7.8.

Recall the fpqc-topology of Definition 1.7.3.

Proposition 1.7.10. Let S be affine. Let X be a torsor in the category ofaffine schemes. Assume that X/S is faithfully flat. Then there are affine group


schemes Gl and Gr operating from the left and right on X, respectively, suchthat the natural maps

Gl ×X → X ×X (g, x) 7→ (gx, x)

X ×Gr → X ×X (x, g′) 7→ (x, xg′)

are isomorphisms.

Moreover, X is a left Gl- and right Gr-torsor with respect to the fpqc-topologyon the category of affine schemes.

Proof. We consider Gl. The arguments for Gr are the same. We define Gl asthe fpqc-sheafification of the presheaf

T 7→ X2(T )/ ∼l

We are going to see below that it is representable by an affine scheme. Themap of presheaves µl defines a multiplication on Gl. It is associative as it isassociative on the presheaf level.

We construct the neutral element. Recall that X → S is an fpqc-cover. Thediagonal ∆ : X → X2/ ∼l induces a section e ∈ G(X). It satisfies descentfor the cover X/S by the definition of the equivalence relation ∼l. Hence itdefines an element e ∈ G(S). We claim that it is the neutral element of G.This can be tested fpqc-locally, e.g., after base change to X. For T/X the setX(T ) is non-empty, hence X2/ ∼l (T ) is a group with neutral element e byProposition 1.7.8.

The inversion map ι exists on X2(T )/ ∼l for T/X, hence it also exists and isthe inverse on G(T ) for T/X. By the sheaf condition this gives a well-definedmap with the correct properties on G.

By the same arguments, the action homomorphism X2(T )/ ∼l ×X(T )→ X(T )defines a left action Gl × X → X. The induced map Gl × X → X × X is anisomorphism because it as an isomorphism on the presheaf level for T/X. Inparticular, X is a left Gl-torsor.

We now turn to representability.

We are going to construct Gl by flat descent with respect to the faithfully flatcover X → S following [BLR, Section 6.1]. In order to avoid confusion, putT = X and Y = X ×X viewed as T -scheme over the second factor. A descentdatum on Y → T consists of the choice of an isomorphism

φ : p∗1Y → p∗2Y

subject to the coycle condition

p∗13φ = p∗23φ p∗12φ

with the obvious notation. We have p∗1Y = Y ×T = X2×X and p∗2Y = T×Y =X ×X2 and use

φ(x1, x2, x3) = (x2, ρ(x1, x2, x3), x3)

1.7. TORSORS 51

where ρ : X2 → X is the structural morphism of X. We have p∗12p∗1Y =

X2 ×X ×X etc. and

p∗12φ(x1, x2, x3, x4) = (x2, ρ(x1, x2, x3), x3, x4)

p∗23φ(x1, x2, x3, x4) = (x1, x3, ρ(x2, x3, x4), x4)

p∗13φ(x1, x2, x3, x4) = (x2, x3ρ(x1, x3, x4), x4)

and the cocyle condition is equivalent to

ρ(ρ(x1, x2, x3), x3, x4) = ρ(x1, x2, x4),

which is an immediate consequence of the properties of a torsor. In the affinecase (that we are in) any descent datum is effective, i.e., induced from a uniquelydetermined S-scheme Gl. In other words, it represents the fpqc-sheaf defined asthe coequalizer of

X2 ×X ⇒ X2

with respect to the projection p1 mapping (x1, x2, x3) to (x1, x2) and p2 φ :X2 ×X → X ×X2 → X2 mapping

(x1, x2, x3) 7→ (x2, ρ(x1, x2, x3), x3) 7→ (ρ(x1, x2, x3), x3)

This is precisely the equivalence relation ∼l. Hence

Gl = X2/ ∼l

as fpqc-sheaves.

Remark 1.7.11. If S is the spectrum of a field, then the flatness assumption isalways satisfied. In general, some kind of assumption is needed, as the followingexample shows. Let S be the spectrum of a discrete valuation ring with closedpoint s. Let G be an algebraic group over s and X = G the trivial torsor definedby G. In particular, we have the structure map

X ×s X ×s X → X.

We now view X as an S-scheme. Note that

X ×S X ×S X = X ×s X ×s X

hence X is also a torsor over S in the sense of Definition 1.7.9. However, itis not a torsor with respect to the fpqc (or any other reasonable Grothendiecktopology) as X(T ) is empty for all T → S surjective.


Chapter 2

Singular Cohomology

In this chapter we give a short introduction to singular cohomology. Manyproperties are only sketched, as this theory is considerably easier than de Rhamcohomology for example.

2.1 Sheaf cohomology

Let X be a topological space. Sometimes, if indicated, X will be the underlyingtopological space of an analytic or algebraic variety also denoted by X. To avoidtechnicalities, X will always be assumed to be a paracompact space, i.e., locallycompact, Hausdorff, and satisfying the second countability axiom.

From now on, let F be a sheaf of abelian groups on X and consider sheafcohomology Hi(X,F) from Section 1.4. Mostly, we will consider the case of theconstant sheaf F = Z. Later we will also consider other constant coefficientsR ⊃ Z, but this will not change the following topological statements.

Definition 2.1.1 (Relative Cohomology). Let A ⊂ X be a closed subset, U =X \A the open complement, i : A → X and j : U → X be the inclusion maps.We define relative cohomology as

Hi(X,A;Z) := Hi(X, j!Z),

where j! is the extension by zero, i.e., the sheafification of the presheaf V 7→ Zfor V ⊂ U and V 7→ 0 else.

Remark 2.1.2 (Functoriality and homotopy invariance). The association

(X,A) 7→ Hi(X,A;Z)

is a contravariant functor from pairs of topological spaces to abelian groups. Inparticular, for every continuous map f : (X,A) → (X ′, A′) of pairs, i.e., satis-fying f(A) ⊂ A′, one has a homomorphism f∗ : Hi(X ′, A′;Z) → Hi(X,A;Z).

53

54 CHAPTER 2. SINGULAR COHOMOLOGY

Given two homotopic maps f and g, then the homomorphisms f∗, g∗ are equal.As a consequence, if two pairs (X,A) and (X ′, A′) are homotopy equivalent,then the cohomology groups Hi(X ′, A′;Z) and Hi(X,A;Z) are isomorphic.

Proposition 2.1.3. There is a long exact sequence

· · · → Hi(X,A;Z)→ Hi(X,Z)→ Hi(A,Z)δ→Hi+1(X,A;Z)→ · · ·

Proof. This follows from the exact sequence of sheaves

0→ j!Z→ Z→ i∗Z→ 0.

Note that by our definition of cones, see section 1.3, one has a quasi-isomorphismj!Z = Cone(Z→ i∗Z)[−1]. For Nori motives we need a version for triples, whichcan be proved using iterated cones by the same method:

Corollary 2.1.4. Let X ⊃ A ⊃ B be a triple of relative closed subsets. Thenthere is a long exact sequence

· · · → Hi(X,A;Z)→ Hi(X,B;Z)→ Hi(A,B;Z)δ→Hi+1(X,A;Z)→ · · ·

Here, δ is the connecting homomorphism, which in the cone picture is explainedin Section 1.3.

Proposition 2.1.5 (Mayer-Vietoris). Let U, V be an open cover of X. LetA ⊂ X be closed. Then there is a natural long exact sequence

· · · → Hi(X,A;Z)→ HidR(U,U ∩A;Z)⊕Hi(V, V ∩A;Z)

→ Hi(U ∩ V,U ∩ V ∩A;Z)→ Hi+1(X,A;Z)→ · · ·

Proof. Pairs (U, V ) of open subsets form an excisive couple in the sense of [Spa,pg. 188], and therefore the Mayer-Vietoris property holds by [Spa, pg. 189-190].

Theorem 2.1.6 (Proper base change). Let π : X → Y be proper (i.e., thepreimage of a compact subset is compact). Let F be a sheaf on X. Then thestalk in y ∈ Y is computed as

(Riπ∗F)y = Hi(Xy,F|Xy ).

Proof. See [KS] Proposition 2.6.7. As π is proper, we have Rπ∗ = Rπ!.

Now we list some properties of the sheaf cohomology of algebraic varieties overa field k → C. As usual, we will not distinguish in notation between a varietyX and the topological space X(C). The first property is:

2.1. SHEAF COHOMOLOGY 55

Proposition 2.1.7 (Excision, or abstract blow-up). Let f : (X ′, D′)→ (X,D)be a proper, surjective morphism of algebraic varieties over C, which induces anisomorphism F : X ′ \D′ → X \D. Then

f∗ : H∗(X,D;Z) ∼= H∗(X ′, D′;Z).

Proof. This fact goes back to A. Aeppli [Ae]. It is a special case of proper-basechange: Let j : U → X be the complement of D and j′ : U → X ′ its inclusioninto X ′. For all x ∈ X, we have

Riπ∗j′!Z = Hi(Xx, j

′!Z|X′x).

For x ∈ U , the fibre is one point. It has no higher cohomology. For x ∈ D, therestriction of j′!Z to X ′x is zero. Together this means

Rπ∗j′!Z = j!Z.

The statement follows from the Leray spectral sequence.

We will later prove a slightly more general proper base change theorem forsingular cohomology, see Theorem 2.5.12.

The second property is:

Proposition 2.1.8 (Gysin isomorphism, localization, weak purity). Let X bean irreducible variety of dimension n over k, and Z a closed subvariety of purecodimension r. Then there is an exact sequence

· · · → HiZ(X,Z)→ Hi(X,Z)→ Hi(X \ Z,Z)→ Hi+1

Z (X,Z)→ · · ·

where HiZ(X,Z) is cohomology with supports in Z, defined as the hypercohomol-

ogy of Cone(ZX → ZX\U )[−1].

If, moreover, X and Z are both smooth, then one has the Gysin isomorphism

HiZ(X,Z) ∼= Hi−2r(Z,Z).

In particular, one has weak purity:

HiZ(X,Z) = 0 for i < 2r,

and H2rZ (X,Z) = H0(Z,Z) is free of rank the number of components of Z.

Proof. See [Pa, Sect. 2] for this statement and an axiomatic treatment withmore general properties and examples of cohomology theories.


2.2 Singular (co)homology

Let X be a topological space (same general assumptions as in section 2.1). Thedefinition of singular homology and cohomology uses topological simplexes.

Definition 2.2.1. The topological n-simplex ∆n is defined as

∆n := (t0, ..., tn) |n∑i=0

ti = 1, ti ≥ 0 .

∆n has natural codimension one faces defined by ti = 0.

Singular (co)homology is defined by looking at all possible continuous mapsfrom simplices to X.

Definition 2.2.2. A singular n-simplex σ is a continuous map

f : ∆n → X.

In the case where X is a differentiable manifold, a singular simplex σ is calleddifferentiable, if the map f can be extended to a C∞-map from a neighbourhoodof ∆n ⊂ Rn+1 to X. The group of singular n-chains is the free abelian group

Sn(X) := Z[f : ∆n → X | f singular chain ].

In a similar way, we denote by S∞n (X) the free abelian group of differentiablechains. The boundary map ∂n : Sn(X)→ Sn−1(X) is defined as

∂n(f) :=

n∑i=0

(−1)if |ti=0.

The group of singular n-cochains is the free abelian group

Sn(X) := HomZ(Sn(X),Z).

The group of differentiable singular n-cochains is the free abelian group

Sn(X) := HomZ(S∞n (X),Z).

The adjoint of ∂n+1 defines the boundary map

dn : Sn(X)→ Sn+1(X).

Lemma 2.2.3. One has ∂n−1∂n = 0 and dn+1dn = 0, i.e., the groups S•(X)and S•(X) define complexes of abelian groups.

The proof is left to the reader as an exercise.

2.3. SIMPLICIAL COHOMOLOGY 57

Definition 2.2.4. Singular homology and cohomology with values in Z is de-fined as

Hising(X,Z) := Hi(S•(X), d•), H

singi (X,Z) := Hi(S•(X), ∂•) .

In a similar way, we define (forX a manifold) the differentiable singular (co)homologyas

Hising,∞(X,Z) := Hi(S•∞(X), d•), H

sing,∞i (X,Z) := Hi(S

∞• (X), ∂•) .

Theorem 2.2.5. Assume that X is a locally contractible topological space, i.e.,every point has an open contractible neighborhood. In this case, singular coho-mology Hi

sing(X,Z) agrees with sheaf cohomology Hi(X,Z) with coefficients inZ. If Y is a differentiable manifold, differentiable singular (co)homology agreeswith singular (co)homology.

Proof. Let Sn be the sheaf associated to the presheaf U 7→ Sn(U). One showsthat Z → S• is a fine resolution of the constant sheaf Z [Wa, pg. 196]. Inthe proof it is used that X is locally contractible, see [Wa, pg. 194]. If X isa manifold, differentiable cochains also define a fine resolution [Wa, pg. 196].Therefore, the inclusion of complexes S∞• (X) → S•(X) induces isomorphisms

Hising,∞(X,Z) ∼= Hi

sing(X,Z) and Hsing,∞i (X,Z) ∼= Hsing

i (X,Z) .

Of course, topological manifolds satisfy the assumption of the theorem.

2.3 Simplicial cohomology

In this section we want to introduce simplicial (co)homology and its relation tosingular (co)homology. Simplicial (co)homology can be defined for topologicalspaces with an underlying combinatorial structure.

In the literature there are various notions of such spaces. In increasing order ofgenerality, these are: (geometric) simplicial complexes and topological realiza-tions of abstract simplicial complexes, of ∆-complexes (sometimes also calledsemi-simplicial complexes), and of simplicial sets. A good reference with a dis-cussion of various definitions is the book by Hatcher [Hat], or the introductorypaper [Fri] by Friedman.

By construction, such spaces are built from topological simplices ∆n in variousdimensions n, and the faces of each simplex are of the same type. Particularlynice examples are polyhedra, for example a tetrahedron, where the simplicialstructure is obvious.

Geometric simplicial complexes come up more generally in geometric situationsin the triangulations of manifolds with certain conditions. An example is the


case of an analytic space Xan where X is an algebraic variety defined overR. There one can always find a semi-algebraic triangulation by a result ofLojasiewicz, cf. Hironaka [Hi2, p. 170] and Prop. 2.6.8.

In this section, we will think of a simplicial space as the topological realizationof a finite simplicial set:

Definition 2.3.1. Let X• be a finite simplicial set in the sense of Remark 1.5.5.One has the face maps

∂i : Xn → Xn−1 i = 0, . . . , n,

and the degeneracy maps

si : Xn → Xn+1 i = 0, . . . , n.

The topological realization |X•| of X• is defined as

|X•| :=∞∐n=0

Xn ×∆n/ ∼,

where each Xn carries the discrete topology, ∆n is the topological n-simplex,and the equivalence relation is given by the two relations

(x, ∂i(y)) ∼ (∂i(x), y), (x, si(y)) ∼ (si(x), y), x ∈ Xn−1, y ∈ ∆n.

(Note that we denote the face and degeneracy maps for the n-simplex by thesame letters ∂i, si.)

In this way, every finite simplicial set gives rise to a topological space |X•|.It is known that |X•| is a compactly generated CW-complex [Hat, Appendix].In fact, every finite CW-complex is homotopy equivalent to a finite simplicalcomplex of the same dimension by [Hat, Thm. 2C.5]. Thus, our restriction torealizations of finite simplicial sets is not a severe restriction.

The skeleton filtration from Remark 1.5.5 defines a filtration of |X•|

|sq0X•| ⊆ |sq1X•| ⊆ · · · ⊆ |sqNX•| = |X•|

by closed subspaces, if Xn is empty for n > N .

There is finite number of simplices in each degree n. Associated to each of themis a continuous map σ : ∆n → |X•|. We denote the free abelian group of allsuch σ of degree n by C∆

n (X•)

∂n : C∆n (X•)→ C∆

n−1(X•)

are given by alternating restriction maps to faces, as in the case of singularhomology. Note that the vertices of each simplex are ordered, so that this iswell-defined.

2.3. SIMPLICIAL COHOMOLOGY 59

Definition 2.3.2. Simplicial homology of the topological space X = |X•| isdefined as

Hsimpln (X,Z) := Hn(C∆

∗ (X•), ∂∗),

and simplicial cohomology as

Hnsimpl(X;Z) := Hn(C∗∆(X•), d∗),

where Cn∆(X•) = Hom(C∆n (X•),Z) and dn is adjoint to ∂n.

Example 2.3.3. A tetrahedron arises from a simplicial set with four vertices (0-simplices), six edges (1-simplices), and four faces (2-simplices). A computationshows that Hn = Z for i = 0, 2 and zero otherwise (this was a priori clear, sinceit is topologically a sphere).

A torus T 2 can be obtained from a square by identifying opposite sides, called aand b. If we look at the diagonal of the square, we see that there is a simplicialcomplex with one vertex (!), three edges, and two faces. A computation showsthat H1(T 2,Z) = Z⊕ Z as expected, and H0(T 2,Z) = H2(T 2,Z) = Z.

This definition does not depend on the representation of a topological spaceX as the topological realization of a simplicial set, since one can prove thatsimplicial (co)homology coincides with singular (co)homology:

Theorem 2.3.4. Singular and simplicial (co)homology of X are equal.

Proof. (For homology only.) The chain of closed subsets

|sq0X•| ⊆ |sq1X•| ⊆ · · · ⊆ |sqNX•| = |X•|

gives rise to long exact sequences of simplicial homology groups

· · · → Hsimpln (|sqn−1X•|,Z)→ Hsimpl

n (|sqnX•|,Z)→ Hsimpln (|sqn−1X•|, |sqnX•|;Z)→ · · ·

A similar sequence holds for singular homology, and there is a canonical mapC∆n (X)→ Cn(X) from simplicial to singular chains. The result is then proved

by induction on n. We use that the relative complex Cn(|sqn−1X•|, |sqnX•|)has zero differential and is a free abelian group of rank equal to the cardinalityof Xn. Therefore, one concludes by observing a computation of the singular(co)homology of ∆n, i.e., Hi(∆n,Z) = Z for i = 0 and zero otherwise.

In a similar way, one can define the simplicial (co)homology of a pair (X,D) =(|X•|, |D•|), where D• ⊂ X• is a simplicial subobject. The associated chaincomplex is given by the quotient complex C∗(X•)/C∗(D•). The same proof willthen show that the singular and simplicial (co)homology of pairs coincide.

From the definition of the topological realization, we see that X is a CW-complex. In the special case, when X is the topological space underlying anaffine algebraic variety X over C, or more generally a Stein manifold, then onecan show:


Theorem 2.3.5 (Artin vanishing). Let X be an affine variety over C of dimen-sion n. Then Hq(Xan,Z) = 0 for q > n. In fact, Xan is homotopy equivalentto a finite simplicial complex where all cells are of dimension ≤ n.

Proof. The proof was first given by Andreotti and Fraenkel [AF] for Stein man-ifolds. For Stein spaces, i.e., allowing singularities, this is a theorem of Kaup,Narasimhan and Hamm, see [Ham1, Satz 1] and the correction in [Ham2]. Analgebraic proof was given by M. Artin [A, Cor. 3.5, tome 3].

Corollary 2.3.6 (Good topological filtration). Let X be an affine variety overC of dimension n. Then the skeleton filtration of Xan is given by

Xan = Xn ⊃ Xn−1 ⊃ · · · ⊃ X0

where the pairs (Xi, Xi−1) have only cohomology in degree i.

Remark 2.3.7. The Basic Lemma of Nori and Beilinson, see Thm. 2.5.7, showsthat there is even an algebraic variant of this topological skeleton filtration.

Corollary 2.3.8 (Artin vanishing for relative cohomology). Let X be an affinevariety of dimension n and Z ⊂ X a closed subvariety. Then

Hi(Xan, Zan,Z) = 0 for i > n.

Proof. Consider the long exact sequence for relative cohomology and use Artinvanishing for X and Z from Thm.2.3.5.

The following theorem is strongly related to the Artin vanishing theorem.

Theorem 2.3.9 (Lefschetz hyperplane theorem). Let X ⊂ PNC be an integralprojective variety of dimension n, and H ⊂ PNC a hyperplane section such thatH∩X contains the singularity set Xsing of X. Then the inclusion H∩X ⊂ X is(n− 1)-connected. In particular, one has Hq(X,Z) = Hq(X ∩H,Z) for q ≤ n.

Proof. See for example [AF].

2.4 Kunneth formula and Poincare duality

Assume that we have given two topological spaces X and Y , and two closedsubsets j : A → X, and j′ : C → Y . By the above, we have

H∗(X,A;Z) = H∗(X, j!Z)

andH∗(Y,C;Z) = H∗(Y, j′!Z) .

The relative cohomology group

H∗(X × Y,X × C ∪A× Y ;Z)

2.4. KUNNETH FORMULA AND POINCARE DUALITY 61

can be computed as H∗(X × Y, j!Z), where

j : X × C ∪A× Y → X × Y

is the inclusion map. One has j! = j! j′! . Hence, we have a natural exteriorproduct map

Hi(X,A;Z)⊗Hj(Y,C;Z)×−→Hi+j(X × Y,X × C ∪A× Y ;Z).

This is related to the so-called Kunneth formula:

Theorem 2.4.1 (Kunneth formula for pairs). Let A ⊂ X and C ⊂ Y be closedsubsets. The exterior product map induces a natural isomorphism⊕

i+j=n

Hi(X,A;Q)⊗Hj(Y,C;Q)∼=−→Hn(X × Y,X × C ∪A× Y ;Q).

The same result holds with Z-coefficients, provided all cohomology groups of(X,A) and (Y,C) in all degrees are free.

Proof. Using the sheaves of singular cochains, see the proof of Theorem 2.2.5,one has fine resolutions j!Z → F • on X, and j′!Z → G• on Y . The tensorproduct F •G• thus is a fine resolution of j!Z = j!Z j′!Z. Here one uses thatthe tensor product of fine sheaves is fine [Wa, pg. 193]. The cohomology of thetensor product complex F • ⊗G• induces a short exact sequence

0→⊕i+j=n

Hi(X,A;Z)⊗Hj(Y,C;Z)→ Hn(X × Y,X × C ∪A× Y ;Z)

→⊕

i+j=n+1

TorZ1 (Hi(X,A;Z), Hj(Y,C;Z))→ 0

by [God, thm. 5.5.1] or [We, thm. 3.6.3]. If all cohomology groups are free, thelast term vanishes.

Proposition 2.4.2. The Kunneth isomorphism of Theorem 2.4.1 is associativeand graded commutative.

Proof. This is a standard consequence of the definition of the Kunneth isomor-phism from complexes of groups.

In later constructions, we will need a certain compatibility of the exterior prod-uct with coboundary maps. Assume that X ⊃ A ⊃ B and Y ⊃ C are closedsubsets.

Proposition 2.4.3. The diagram involving coboundary maps

Hi(A,B;Z)⊗Hj(Y,C;Z) −−−−→ Hi+j(A× Y,A× C ∪B × Y ;Z)

δ⊗id

y yδHi+1(X,A;Z)⊗Hj(Y,C;Z) −−−−→ Hi+j+1(X × Y,X × C ∪A× Y ;Z)


commutes up to a sign (−1)j. The diagram

Hi(Y,C;Z)⊗Hj(A,B;Z) −−−−→ Hi+j(Y ×A, Y ×B ∪ C ×A;Z)

id⊗δy yδ

Hi(Y,C;Z)⊗Hj+1(X,A;Z) −−−−→ Hi+j+1(Y ×X,Y ×A ∪ C ×X;Z)

commutes (without a sign).

Proof. We indicate the argument, without going into full details. Let F • bea complex computing H•(Y,C;Z) Let G•1 and G•2 be complexes computingH•(A,B;Z) and H•(X,A;Z). Let K•1 and K•2 be the complexes computingcohomology of the corresponding product varieties. Cup product is inducedfrom maps of complexes F •i ⊗G• → K•i . In order to get compatibility with theboundary map, we have to consider the diagram of the form

F1 ⊗G −−−−→ K1y y(F2[1])⊗G −−−−→ K2[1]

However, by Lemma 1.3.6, the complexes (F2[1]) ⊗ G and (F2 ⊗ G)[1] are notequal. We need to introduce the sign (−1)j in bidegree (i, j) to make theidentification and get a commutative diagram.

The argument for the second type of boundary map is the same, but does notneed the introduction of signs by Lemma 1.3.6.

Assume now that X = Y and A = C. Then, j!Z has an algebra structure, andwe obtain the cup product maps:

Hi(X,A;Z)⊗Hj(X,A;Z) −→ Hi+j(X,A;Z)

via the multiplication maps

Hi+j(X ×X, j!Z)→ Hi+j(X, j!Z),

induced byj! = j! j! → j! .

In the case where A = ∅, the cup product induces Poincare duality:

Proposition 2.4.4 (Poincare Duality). Let X be a compact, orientable topo-logical manifold of dimension m. Then the cup product pairing

Hi(X,Q)×Hm−i(X,Q) −→ Hm(X,Q) ∼= Q

is non-degenerate in both factors. With Z-coefficients, the same result holds forthe two left groups modulo torsion.

2.4. KUNNETH FORMULA AND POINCARE DUALITY 63

Proof. We will give a proof of a slightly more general statement in the algebraicsituation below. A proof of the stated theorem can be found in [GH, pg. 53].There it is shown that H2n(X) is torsion-free of rank one, and the cup-productpairing is unimodular modulo torsion, using simplicial cohomology, and therelation between Poincare duality and the dual cell decomposition.

We will now prove a relative version in the algebraic case. It implies the versionabove in the case where A = B = ∅. By abuse of notation, we again do notdistinguish between an algebraic variety over C and its underlying topologicalspace.

Theorem 2.4.5 (Poincare duality for algebraic pairs). Let X be a smooth andproper complex variety of dimension n over C and A,B ⊂ X two normal cross-ing divisors, such that A ∪B is also a normal crossing divisor. Then there is anon-degenerate duality pairing

Hd(X\A,B\(A∩B);Q)×H2n−d(X\B,A\(A∩B);Q) −→ H2n(X,Q) ∼= Q(−n).

Again, with Z-coefficients this is true modulo torsion by unimodularity of thecup-product pairing.

Proof. We give a sheaf theoretic proof using Verdier duality and some formulasand ideas of Beilinson (see [Be1]). Look at the commutative diagram:

U = X \ (A ∪B)Ù−−−−→ X \A

κU

y yκX \B `−−−−→ X.

Assuming A ∪ B is a normal crossing divisor, we want to show first that thenatural map

`!RκU∗QU −→ Rκ∗Ù !QU ,

extending id : QU → QU , is an isomorphism. This is a local computation.We look without loss of generality at a neighborhood of an intersection pointx ∈ A ∩ B, since the computation at other points is even easier. If we work inthe analytic topology, we may choose a polydisk neighborhood D in X aroundx such that D decomposes as

D = DA ×DB

and such that

A ∩D = A0 ×DB , B ∩D = DA ×B0


for some suitable topological spaces A0, B0. Using the same symbols for themaps as in the above diagram, the situation looks locally like

(DA \A0)× (DB \B0)Ù−−−−→ (DA \A0)×DB

κU

y yκDA × (DB \B0)

`−−−−→ D = DA ×DB .

Using the Kunneth formula, one concludes that both sides `!RκU∗QU andRκ∗Ù !QU are isomorphic to

RκU∗QDA\A0⊗ `!QDB\B0

near the point x, and the natural map provides an isomorphism.

Now, one has

Hd(X \A,B \ (A ∩B));Q) = Hd(X, `!κU∗QU ),

(using that the maps involved are affine), and

H2n−d(X \B,A \ (A ∩B));Q) = H2n−d(X,κ!Ù∗QU ).

We have to show that there is a perfect pairing

Hd(X \A,B \ (A ∩B));Q)×H2n−d(X \B,A \ (A ∩B));Q)→ Q(−n).

However, by Verdier duality, we have a perfect pairing

H2n−d(X \B,A \ (A ∩B));Q)∨ = H2n−d(X,κ!Ù∗QU )∨

= H−d(X,κ!Ù∗DQU )(−n)

= H−d(X,D(κ∗Ù !QU ))(−n)

= Hd(X,κ∗Ù !QU )(−n)

= Hd(X, `!κU∗QU )(−n)

= Hd(X \A,B \ (A ∩B));Q).

Remark 2.4.6. The normal crossing condition is necessary, as one can see inthe example of X = P2, where A consists of two distinct lines meeting in apoint, and B a line going through the same point.

2.5 Basic Lemma

In this section we prove the basic lemma of Nori [N, N1, N2], a topologicalresult, which was also known to Beilinson [Be1] and Vilonen (unpublished). Letk ⊂ C be a subfield. The proof of Beilinson works more generally in positivecharacteristics as we will see below.

2.5. BASIC LEMMA 65

Convention 2.5.1. We fix an embedding k → C. All sheaves and all cohomol-ogy groups in the following section are to be understood in the analytic topologyon X(C).

Theorem 2.5.2 (Basic Lemma I). Let k ⊂ C. Let X be an affine variety over kof dimension n and W ⊂ X be a Zariski closed subset with dim(W ) < n. Thenthere exists a Zariski closed subset Z ⊃W defined over k with dim(Z) < n and

Hq(X,Z;Z) = 0, for q 6= n

and, moreover, the cohomology group Hn(X,Z;Z) is a free Z-module.

We formulate the Lemma for coefficients in Z, but by the universal coefficienttheorem [We, thm. 3.6.4] it will hold with other coefficients as well.

Example 2.5.3. There is an example where there is an easy way to obtain Z.Assume that X is of the form X \H for some smooth projective X and a hyper-plane H. Let W = ∅. Then take another hyperplane section H ′ meeting X andH transversally. Then Z := H ′∩X will have the property that Hq(X,Z;Z) = 0for q 6= n by the Lefschetz hyperplane theorem, see Thm. 2.3.9. This argumentwill be generalized in two of the proofs below.

An inductive application of this Basic Lemma in the case Z = ∅ yields a filtrationof X by closed subsets

X = Xn ⊃ Xn−1 ⊃ · · · ⊃ X0 ⊃ X−1 = ∅

with dim(Xi) = i such that the complex of free Z-modules

· · · δi−1−→Hi(Xi, Xi−1)δi−→Hi+1(Xi+1, Xi)

δi+1−→· · · ,

where the maps δ• arise from the coboundary in the long exact sequence asso-ciated to the triples Xi+1 ⊃ Xi ⊃ Xi−1, computes the cohomology of X.

Remark 2.5.4. This means that we can understand this filtration as algebraicanalogue of the skeletal filtraton of simplicial complexes, see Corollary 2.3.6.Note that the filtration is not only algebraic, but even defined over the basefield k.

The Basic Lemma is deduced from the following variant, which was also knownto Beilinson [Be1]. To state it, we need the notion of a (weakly) constructiblesheaf.

Definition 2.5.5. A sheaf of abelian groups on a variety X over k is weaklyconstructible, if there is a stratification of X into a disjoint union of finitely manyZariski locally closed subsets Yi defined over k, and such that the restriction ofF to Yi is locally constant. It is called constructible if, in addition, the stalks ofF are finitely generated abelian groups.


Remark 2.5.6. This combination of sheaves in the analytic topology and strataalgebraic and defined over k, is not very much discussed in the literature. In fact,the formalism works in the same way as with algebraic strata over k. What weneed are enough Whitney stratifications algebraic over k. That this is possiblecan be deduced from [Tei, Theoreme 1.2 p. 455] (characterization of Whitneystratifications) and [Tei, Proposition 2.1] (Whitney stratifications are generic).

We will also need some basic facts about sheaf cohomology. If j : U → X isa Zariski open subset with closed complement i : W → X and F a sheaf ofabelian groups on X, then there is an exact sequence of sheaves

0→ j!j∗F → F → i∗i

∗F → 0.

In addition, for F the constant sheaf Z, one has Hq(X, j!j∗F ) = Hq(X,W ;Z)

and Hq(X, i∗i∗F ) = Hq(W,Z), see section 2.1.

Theorem 2.5.7 (Basic Lemma II). Let X be an affine variety over k of dimen-sion n and F be a weakly constructible sheaf on X. Then there exists a Zariskiopen subset j : U → X such the following three properties hold:

1. dim(X \ U) < n.

2. Hq(X,F ′) = 0 for q 6= n, where F ′ := j!j∗F ⊂ F .

3. If F is constructible, then Hn(X,F ′) is finitely generated.

4. If the stalks of F are torsion free, then Hn(X,F ′) is torsion free.

Version II of the Lemma implies version I. Let V = X \W with open immer-sion h : V → X, and the sheaf F = h!h

∗Z on X. Version II for F gives an opensubset ` : U → X such that the sheaf F ′ = `!`

∗F has non-vanishing cohomologyonly in degree n. Let W ′ = X \ U . Since F was zero on W , we have that F ′ iszero on Z = W ∪W ′ and it is the constant sheaf on X \ Z, i.e., F ′ = j!j

∗F forj : X \Z → X. In particular, F ′ computes the relative cohomology Hq(X,Z;Z)and it vanishes for q 6= n. Freeness follows from property (3).

We will give two proofs of the Basic Lemma II below.

2.5.1 Direct proof of Basic Lemma I

We start by giving a direct proof of Basic Lemma I. It was given by Nori in theunpublished notes [N1]. Close inspection shows that it is actually a variant ofBeilinson’s argument in this very special case.

Lemma 2.5.8. Let X be affine, W ⊂ X closed. Then there exist

1. X smooth projective;

2.5. BASIC LEMMA 67

2. D0, D∞ ⊂ X closed such that D0∪D∞ is a simple normal crossings divisorand X \D0 is affine;

3. π : X \ D∞ → X proper surjective, an isomorphism outside of D0 suchthat Y := π(D0 \D∞ ∩D0) contains W and π−1(Y ) = D0 \D∞ ∩D0.

Proof. We may assume without loss of generality that X \W is smooth. LetX be a projective closure of X and W the closure of W in X. By resolutionof singularities, there is X → X proper surjective and an isomorphism aboveX \W such that X is smooth. Let D∞ ⊂ X be the complement of the preimageof X. Let W be the closure of the preimage of X. By resolution of singularities,we can also assume that W ∪D∞ is a divisor with normal crossings.

Note that X and hence also X are projective. We choose a generic hyperplaneH such that W ∪D∞ ∪ H is a divisor with normal crossings. This is possibleas the ground field k is infinite and the condition is satisfied in a Zariski opensubset of the space of hyperplane sections. We put D0 = H ∪ W . As H isa hyperplane section, it is an ample divisor. Therefore, D0 = H ∪ W is thesupport of the ample divisor H + mW for m sufficiently large [Ha2, Exer. II7.5(b)]. Hence X \ D0 is affine, as the complement of an ample divisor in aprojective variety is affine.

Proof of Basic Lemma I. We use the varieties constructed in the last lemma.We claim that Y has the right properties if the coefficients form an arbitraryfield K. We have Y ⊃ W . From Artin vanishing, see Corollary 2.3.8, weimmediately have vanishing of Hi(X,Y ;K) for i > n.

By excision (see Proposition 2.1.7)

Hi(X,Y ;K) = Hi(X \D∞, D0 \D0 ∩D∞;K).

By Poincare duality for pairs (see Theorem 2.4.5), it is dual to

H2n−i(X \D0, D∞ \D0 ∩D∞;K).

The variety X \D0 is affine. Hence by Artin vanishing, the cohomology groupvanishes for all i 6= n and any coefficient field K.

It remains to treat the case of integral coefficients. Let i be the smallest indexsuch that Hi(X,Y ;Z) is non-zero. By Artin vanishing for Z-coefficients 2.3.5,we have i ≤ n.

If i < n, then the group Hi(X,Y ;Z) has to be torsion because the cohomologyvanishes with Q-coefficients. By the universal coefficient theorem [We, thm.3.6.4]

Hi−1(X,Y ;Fp) = TorZ1 (Hi(X,Y ;Z),Fp) ,

which implies that Hi−1(X,Y ;Fp) is non-trivial for the occuring torsion primes.This is a contradiction to the vanishing for K = Fp. Hence i = n. The sameargument shows that Hn(X,Y ;Z) is torsion-free.


2.5.2 Nori’s proof of Basic Lemma II

We now present the proof of the stronger Basic Lemma II published by Nori in[N2].

We start with a couple of lemmas on weakly constructible sheaves.

Lemma 2.5.9. Let 0→ F1 → F2 → F3 → 0 be a short exact sequence of sheaveson X with F1, F3 (weakly) constructible. Then F2 is (weakly) constructible.

Proof. By assumption, there are stratifications of X such that F1 and F3 becomelocally constant, respectively. We take a common refinement. We replace X byone of the strata and are now in the situation that F1 and F3 are locally constant.Then F3 is also locally constant. Indeed, by passing to a suitable open cover(in the analytic topology), F1 and F3 become even constant. If V ⊂ U is aninclusion of open connected subsets, then the restrictions F1(U) → F1(V ) andF3(U) → F3(U) are isomorphisms. This implies the same statement for F2,because H1(U,F1) = H1(V, F1) = 0, as constant sheaves do not have highercohomology.

Lemma 2.5.10. The notion of (weak) constructibility is stable under j! for jan open immersion and π∗ for π finite.

Proof. The assertion of j! is obvious, same as for i∗ for closed immersions.

Now assume π : X → Y is finite and in addition etale. Let F be (weakly)constructible on X. Let X0, . . . , Xn ⊂ X be the stratification such that F |Xi islocally constant. Let Yi be the image of Xi. These are locally closed subvarietiesof Y because π is closed and open. We refine them into a stratification of Y .As π is finite etale, it is locally in the analytic topology of the form I ×B withI finite and B ⊂ Y (C) an open in the analytic topology. Obviously π∗F |B islocally constant on the strata we have defined.

Now let π be finite. As we have already discussed closed immersion, it sufficesto assume π surjective. There is an open dense subscheme U ⊂ Y such π isetale above U . Let U ′ = π−1(U), Z = Y \U and Z ′ = X \U ′. We consider theexact sequence on X

0→ jU ′!j∗U ′F → F → iZ′∗i

∗Z′F → 0.

As π is finite, the functor π∗ is exact and hence

0→ π∗jU ′!j∗U ′F → π∗F → π∗iZ′∗i

∗Z′F → 0.

By Lemma 2.5.9, it suffices to consider the outer terms. We have

π∗jU ′!j∗U ′F = jU∗π|U ′∗j∗U ′F,

and this is (weakly) constructible by the etale case and the assertion on openimmersions. We also have

π∗iZ′∗i∗Z′F = iZ∗π|Z′∗i∗Z′F,

2.5. BASIC LEMMA 69

and this is (weakly) constructible by noetherian induction and the case of closedimmersions.

Nori’s proof of Basic Lemma II. The argument will show a more precise versionof property (3): There exists a finite subset E ⊂ U(C) such that Hdim(X)(X,F ′)is isomorphic to a direct sum ⊕xFx of stalks of F at points of E.

Let n := dim(X). In the first step, we reduce to X = An. We use Noethernormalization to obtain a finite morphism π : X → An. By Lemma 2.5.10, thesheaf π∗F is (weakly) constructible.

Let then v : V → An be a Zariski open set with the property that F ′ := v!v∗π∗F

satisfies the Basic Lemma II on An. Let U := π−1(V )j→X be the preimage in

X. One has an equality of sheaves:

π∗j!j∗F = v!v

∗π∗F.

Therefore, Hq(X, j!j∗F ) = Hq(An, v!v

∗π∗F ) and the latter vanishes for q ≤ n.The formula for the n-cohomology on An implies the one on X.

So let us now assume that F is weakly constructible on X = An. We argue byinduction on n and all F . The case n = 0 is trivial.

By replacing F by j!j∗F for an appropriate open j : U → An, we may assume

that F is locally constant on U and that An \ U = V (f). By Noether normal-ization or its proof, there is a surjective projection map π : An → An−1 suchthat π|V (f) : V (f)→ An−1 is surjective and finite.

We will see in Lemma 2.5.11 that Rqπ∗F = 0 for q 6= 1 and R1π∗F is weaklyconstructible. The Leray spectral sequence now gives that

Hq(An, F ) = Hq−1(An−1, R1π∗F ).

In the induction procedure, we apply the Basic Lemma II to R1π∗F on An−1.By induction, there exists a Zariski open h : V → An−1 such that h!h

∗R1π∗Fhas cohomology only in degree n− 1. Let U := π−1(V ) and j : U → An be theinclusion. Then j!j

∗F has cohomology only in degree n. The explicit descriptionof cohomology in degree n follows from the description of the stalks of R1π∗Fin the proof of Lemma 2.5.11.

Lemma 2.5.11. Let π be as in the above proof. Then Rqπ∗F = 0 for q 6= 1and R1π∗F is weakly constructible.

Proof. This is a standard fact, but Nori gives a direct proof.

The stalk of Rqπ∗F at y ∈ An−1 is given by Hq(y × A1, Fy×A1) by thevariation of proper base change in Theorem 2.5.12 below.

Let, more generally, G be a sheaf on A1 which is locally constant outside a finite,non-empty set S. Let T be a tree in A1(C) with vertex set S. Then the restric-tion map to the tree defines a retraction isomorphism Hq(A1, G) ∼= Hq(T,GT )


for all q ≥ 0. Using Cech cohomology, we can compute that Hq(T,GT ): Foreach vertex v ∈ S, let Us be the star of half edges of length more than half thelength of all outgoing edges at the vertex s. Then Ua and Ub only intersect ifthe vertices a and b have a common edge e = e(a, b). The intersection Ua ∩ Ubis contractible and contains the center t(e) of the edge e. There are no tripleintersections. Hence Hq(T,GT ) = 0 for q ≥ 2. Since G is zero on S, Us issimply connected, and G is locally constant, G(Us) = 0 for all s. Therefore alsoH0(T,GT ) = 0 and H1(T,GT ) is isomorphic to

⊕eGt(e).

This implies already that Rqπ∗F = 0 for q 6= 1.

To show that R1π∗F is weakly constructible, means to show that it is locallyconstant on some stratification. We see that the stalks (R1π∗F )y depend onlyon the set of points in y×A1 = π−1(y) where Fy×A1 vanishes. But the setsof points where the vanishing set has the same degree (cardinality) defines asuitable stratification. Note that the stratification only depends on the branch-ing behaviour of V (f)→ An−1, hence the stratification is algebraic and definedover k.

Theorem 2.5.12 (Variation of Proper Base Change). Let π : X → Y be acontinuous map between locally compact, locally contractible topological spaceswhich is a fiber bundle and let G be a sheaf on X. Assume W ⊂ X is closedand such that G is locally constant on X \W and π restricted to W is proper.Then (Rqπ∗G)y ∼= Hq(π−1(y), Gπ−1(y)) for all q and all y ∈ Y .

Proof. The statement is local on Y , so we may assume that X = T×Y is a prod-uct with π the projection. Since Y is locally compact and locally contractible,we may assume that Y is compact by passing to a compact neighbourhood of y.As W → Y is proper, this implies that W is compact. By enlarging W , we mayassume that W = K × Y is a product of compact sets for some compact subsetK ⊂ X. Since Y is locally contractible, we replace Y be a contractible neigh-bourhood. (We may loose compactness, but this does not matter anymore.) Leti : K × Y → X be the inclusion and and j : (T \K)× Y → X the complement.

Look at the exact sequence

0→ j!G(T\K)×Y → G→ i∗GK×Y → 0.

The result holds for GK×Y by the usual proper base change.

Since Y is contractible, we may assume that G(T\K)×Y is the pull-back of con-stant sheaf on T \K. Now the result for j!G(T\K)×Y follows from the Kunnethformula.

2.5.3 Beilinson’s proof of Basic Lemma II

We follow Beilinson [Be1] Proof 3.3.1. His proof is even more general, as hedoes not assume X to be affine. Note that Beilinson’s proof is in the setting

2.5. BASIC LEMMA 71

of etale sheaves, independent of the characteristic of the ground field. We havetranslated it to weakly constructible sheaves. The argument is intrinsicallyabout perverse sheaves, even though we have downplayed their use as far aspossible. For a complete development of the theory of perverse sheaves in the(weakly) constructible setting see Schurmann’s monograph [Schu].

Let X be affine reduced of dimension n over a field k ⊂ C. Let F be a (weakly)constructible sheaf on X. We choose a projective compactification κ : X → Xsuch that κ is an affine morphism. Let W be a divisor on X such that F is alocally constant sheaf on h : X \W → X and X \W is smooth. Then defineM := h!h

∗F .

Let H ⊂ X be a generic hyperplane. (We will see in the proof of Lemma 2.5.13below what the conditions on H are.) Let H = X ∩ H be the hyperplane in X.

We denote by V = X \H the complement and by ` : V → X the open inclusion.Furthermore, let κV : V ∩X → V and `X : V ∩X → X be the open inclusionmaps, and i : H → X and iX : H → X the closed immersions. We setU := X \ (W ∪H) and consider the open inclusion j : U → X with complementZ = W ∪ (H ∩X). Let MV ∩X be the restriction of M to V ∩X. Summarizing,we have a commutative diagram

UyjV ∩X `X−−−−→ X

iX←−−−− H

κV

y yκ yκV

`−−−−→ Xi←−−−− H.

Lemma 2.5.13. For generic H in the above set-up, there is an isomorphism

`!`∗Rκ∗M

∼=−→ Rκ∗`X∗MV ∩X

extending naturally id : MV ∩X →MV ∩X .

Proof. We consider the map of distinguished triangles

`!`∗Rκ∗M −−−−→ Rκ∗M −−−−→ i∗i

∗Rκ∗My id

y yRκ∗`X!MV ∩X −−−−→ Rκ∗M −−−−→ i∗Rκ∗i

∗XM

(the existence of the arrows follows from standard adjunctions together withproper base change in the simple form κ∗`! = `X!κ

∗V and κ∗i∗ = iX∗κ

∗, respec-tively).

Hence it is sufficient to prove that

i∗Rκ∗M∼=−→Rκ∗i∗XM. (2.1)


To prove this, we make a base change to the universal hyperplane section. Indetail: Let P be the space of hyperplanes in X. Let

HP → P

be the universal family. It comes with a natural map

iP : H → X.

Let H be the preimage of X. By [Gro2, pg. 9] and [Jo, Thm. 6.10] there is adense Zariski open subset T ⊂ P such that the induced map

iT : HT → X × T −→ X

is smooth.

We apply smooth base change in the square

HTiX,T−−−−→ X

κT

y yκHT

iT−−−−→ X

and obtain a quasi-isomorphism

i∗TRκ∗M∼=−→RκT∗i∗X,TM

of complexes of sheaves on HT .

We specialize to some t ∈ T (k) and get a hyperplane t : H ⊂ HT to which werestrict. The left hand side turns into i∗Rκ∗M .

The right hand side turns into

t∗RκT∗i∗X,TM = Rκ∗t

∗X i∗X,TM = Rκi∗XM

by applying the generic base change theorem 2.5.16 to κT over the base T andG = i∗X,TM . This requires to shrink T further.

Putting these equations together, we have verified equation 2.1.

Proof of Basic Lemma II. We keep the notation fixed in this section. By Artinvanishing for constructible sheaves (see Theorem 2.5.14), the groupHi(X, j!j

∗F )vanishes for i > n. It remains to show that Hi(X, j!j

∗F ) vanishes for i < n.We obviously have j!j

∗F = `X!MV ∩X . Therefore,

Hi(X, j!j∗F ) = Hi(X, `X!MV ∩X)

= Hi(X, Rκ∗`X!MV ∩X)

= Hi(X, `!`∗Rκ∗M) by 2.5.13

= Hic(V, (Rκ∗M)V ).

2.5. BASIC LEMMA 73

The last group vanishes for i < n by Artin’s vanishing theorem 2.5.14 for com-pact supports once we have checked that Rκ∗MV [n] is perverse. Recall thatM = h!h

∗F |X\W with F |X\W locally constant sheaf on a smooth variety. HenceF |X\W [n] is perverse. Both h and κ are affine, hence the same is true forRκ∗h!F |X\W by Theorem 2.5.14 3.

If, in addition, F is constructible, then by the same theorem, Rκ∗h!F |X\W isperverse for the second t-structure mentionend in Remark 2.5.15. Hence ourcohomology with compact support is also finitely generated.

If the stalks of F are torsion free, then Rκ∗h!F |X\W is perverse for the thirdt-structure mentionend in Remark 2.5.15. Hence our cohomology with compactsupport is also torsion free.

We now formulate the version of Artin vanishing used in the above proof. IfX is a topological space, and j : X → X an arbitrary compactification, thencohomology with supports with coefficients in a weakly constructible sheaf G isdefined by

Hic(X,G) := Hi(X, j!G).

It follows from proper base change that this is independent of the choice ofcompactification.

Theorem 2.5.14 (Artin vanishing for constructible sheaves). Let X be affineof dimension n.

1. Let G be weakly constructible on X. Then Hq(X,G) = 0 for q > n;

2. Let F• be a perverse sheaf on X for the middle perversity. ThenHqc (X,F•) = 0 for q < 0. More precisely, the complex Rc(X,F•) comput-

ing cohomology with compact support is in D≥0.

3. Let g : U → X be an open immersion and F• a perverse sheaf on U . Thenboth g!F• and Rg∗F• are perverse on X.

Proof. The first two statements are [Schu, Corollary 6.0.4, p. 391]. Note that aweakly constructible sheaf lies in mD≤n(X) in the notation of loc.cit.

The last statement combines the vanishing results for affine morphisms [Schu,Theorem 6.0.4, p. 409] with the standard vanishing for all compactifiable mor-phisms [Schu, Corollary 6.0.5, p. 397] for a morphism of relative dimension0.

Remark 2.5.15. For the notion of a t-structure on a triangulated categoryand perverse sheaves, see the original reference [BBD]. Actually, as explainedin [Schu, Example 6.0.2, p. 377], there are different possible choices for thetriangulated category and inital t-structure D≥0. In each case there is corre-sponding middle perverse t-structure by [Schu, Definition 6.0.3, p. 379]. Thetheorem applies in all of them.


1. The category of complexes of sheaves with weakly constructible cohomol-ogy. It is denoted D(Z) in [Schu, Chapter 6]. The prototype of a perversesheaf is a complex of the form F [n] with F a local system on a smoothvariety of dimension n.

2. The category of bounded complexes of sheaves with constructible coho-mology. It is denoted D(Z)perf in loc.cit. The prototype of a perversesheaf is F [n], with F a local system with finitely generated stalks.

3. In both cases, we can also use the t-structure based on +D≤0, the subcate-gory of complexes in positive degrees with H0 torsion free. The prototypeof a perverse sheaf in this case is F [n], with F a local system with torsionfree stalks.

Theorem 2.5.16 (Generic base change). Let S be of finite type over k, f :X → Y a morphism of S-varieties. Let F be a (weakly) constructible sheaf onX. Then there is a dense open subset U ⊂ S such that:

1. over U , the sheaves Rif∗F are (weakly) constructible and almost all van-ish;

2. the formation of Rif∗F is compatible with any base change S′ → U ⊂ S.

This is the analogue of [SGA 4 1/2, Theoreme 1.9 in sect. Thm. finitude],which is for constructible etale sheaves in the etale setting.

Proof. The case S = Y was treated by Arapura, see [Ara, Theorem 3.1.10]. Weexplain the reduction to this case, using the same arguments as in the etale case.

All schemes can be assumed reduced.

Using Nagata, we can factor f as a composition of an open immersion anda proper map. The assertion holds for the latter by the proper base changetheorem, hence it suffices to consider open immersions.

As the question is local on Y , we may assume that it is affine over S. We canthen cover X by affines. Using the hypercohomology spectral sequence for thecovering, we may reduce to the case X affine. In this case (X and Y affine, fan open immersion) we argue by induction on the dimension of the generic fibreof X → S.

If n = 0, then, at least after shrinking S, we are in the situation where f is theinclusion of a connected component and the assertion is trivial.

We now assume the case n−1. We embed Y into AmS and consider the coordinateprojections pi : Y → A1

S . We apply the inductive hypothesis to the map f overA1S . Hence there is an open dense Ui ⊂ A1

S such that the conclusion is validover p−1Ui. Hence the conclusion is valid over their union, i.e., outside a closedsubvariety Y1 ⊂ Y finite over S. By shrinking S, we may assume that it is finiteetale.

2.6. TRIANGULATION OF ALGEBRAIC VARIETIES 75

We fix the notation in the resulting diagram as follows:

Xf //

a

Y

b

Y1ioo

b1~~S

Let j be the open complement of i. We have checked that j∗Rf∗G is (weakly)constructible and compatible with any base change. We apply Rb∗ to the tri-angle defined by the sequence

j!j∗Rf∗G → Rf∗G → i∗i

∗Rf∗G

and obtainRb∗j!j

∗Rf∗G → Ra∗G → b1∗i∗Rf∗G.

The first two terms are (after shrinking of S) (weakly) constructible by theprevious considerations and the case S = Y . We also obtain that they arecompatible with any base change. Hence the same is true for the third term.As b1 is finite etale this also implies that i∗Rf∗G is (weakly) constructible andcompatible with base change. (Indeed, this follows because a direct sum ofsheaves is constant if and only if every summand is constant.) The same is truefor j!j

∗Rf∗G by the previous considerations and base change for j!. Hence theconclusion also holds for the middle term of the first triangle and we are done.

2.6 Triangulation of Algebraic Varieties

If X is a variety defined over Q, we may ask whether any singular homology classγ ∈ Hsing

• (Xan;Q) can be represented by an object described by polynomials.This is indeed the case: for a precise statement we need several definitions. Theresult will be formulated in Proposition 2.6.8.

This section follows closely the Diploma thesis of Benjamin Friedrich, see [Fr].The results are due to him.

We work over k = Q, i.e., the integral closure of Q in R. Note that Q is a field.

In this section, we use X to denote a variety over Q, and Xan for the associatedanalytic space over C (cf. Subsection 1.2).

2.6.1 Semi-algebraic Sets

Definition 2.6.1 ([Hi2, Def. 1.1., p.166]). A subset of Rn is said to be Q-semi-algebraic, if it is of the form

x ∈ Rn|f(x) ≥ 0


for some polynomial f ∈ Q[x1, . . . , xn], or can be obtained from sets of this formin a finite number of steps, where each step consists of one of the following basicoperations:

1. complementary set,

2. finite intersection,

3. finite union.

We need also a definition for maps:

Definition 2.6.2 (Q-semi-algebraic map [Hi2, p. 168]). A continuous map

f between Q-semi-algebraic sets A ⊆ Rn and B ⊆ Rm is said to be Q-semi-algebraic, if its graph

Γf :=(a, f(a)

)| a ∈ A

⊆ A×B ⊆ Rn+m

is Q-semi-algebraic.

Example 2.6.3. Any polynomial map

f : A −→ B

(a1, . . . , an) 7→ (f1(a1, . . . , an), . . . , fm(a1, . . . , an))

between Q-semi-algebraic sets A ⊆ Rn and B ⊆ Rm with fi ∈ Q[x1, . . . , xn] for

i = 1, . . . ,m is Q-semi-algebraic, since it is continuous and its graph Γf ⊆ Rn+m

is cut out from A×B by the polynomials

yi − fi(x1, . . . , xn) ∈ Q[x1, . . . , xn, y1, . . . , ym] for i = 1, . . . ,m. (2.2)

We can even allow f to be a rational map with rational component functions

fi ∈ Q(x1, . . . , xn), i = 1, . . . ,m

as long as none of the denominators of the fi vanish at a point of A. Theargument remains the same except that the expression (2.2) has to be multipliedby the denominator of fi.

Fact 2.6.4 ([Hi2, Prop. II, p. 167], [Sb, Thm. 3, p. 370]).

By a result of Seidenberg-Tarski, the image (respectively preimage) of a Q-semi-

algebraic set under a Q-semi-algebraic map is again Q-semi-algebraic.

As the name suggests, any algebraic set should be in particular Q-semi-algebraic.

Lemma 2.6.5. Let X be a quasi-projective algebraic variety defined over Q.Then we can regard the complex analytic space Xan associated to the base changeXC = X ×Q C as a bounded Q-semi-algebraic subset

Xan ⊆ RN (2.3)

for some N . Moreover, if f : X → Y is a morphism of varieties defined overQ, we can consider fan : Xan → Y an as a Q-semi-algebraic map.


Remark 2.6.6. We will mostly need the case when X is even affine. ThenX ⊂ Cn is defined by polynomial equations with coefficients in Q. We identifyCn ∼= R2n and rewrite the equations for the real and imaginary part. Hence Xis obviously Q-semialgebraic. In the lemma, we will show in addition that Xcan be embedded as a bounded Q-semialgebraic set.

Proof of Lemma 2.6.5. First step X = PnQ

: Consider

• PnC = (PnQ×QC)an with homogenous coordinates x0, . . . , xn, which we split

as xm = am + ibm with am, bm ∈ R in real and imaginary part, and

• RN , N = 2(n+ 1)2, with coordinates ykl, zklk,l=0,...,n.

We define an explicit map

ψ : PnC[x0:...:xn]

−→ RN(y00,z00,...,ynn,znn)

[x0 : . . . : xn] 7→(. . . ,

Rexkxl∑nm=0 |xm|2︸︷︷︸ykl

,Imxkxl∑nm=0 |xm|2︸︷︷︸zkl

, . . .

)

[a0 + ib0 : . . . : an + ibn] 7→(. . . ,

akal + bkbl∑nm=0 a

2m + b2m︸︷︷︸

ykl

,bkal − akbl∑nm=0 a

2m + b2m︸︷︷︸

zkl

, . . .

).

We can understand this map as a section of a natural fibre bundle on PnC. Itstotal space is given by the set E of hermetian (n+ 1)× (n+ 1)-matrices or rank1. The map

φ : E → PnCmaps a linear map M to its image in Cn+1. We get a section of φ by mappinga 1-dimensional subspace L of Cn+1 to the matrix of the orthogonal projectionfrom Cn+1 to L with respect to the standard hermetian product on Cn+1. Wecan describe this section in coordinates. Let (x0, . . . , xn) ∈ Cn+1 be a vectorof length 1. Then an elementary computation shows that M = (xixj)i,j isthe hermetian projector to the line L = C(x0, . . . , xn). Writing the real andimaginary part of the matrix M separately gives us precisely the formula for ψ.In particular, ψ is injective.

Therefore, we can consider PnC via ψ as a subset of RN . It is bounded sinceit is contained in the unit sphere SN−1 ⊂ RN . We claim that ψ(PnC) is also

Q-semi-algebraic. The composition of the projection

π : R2(n+1) \ (0, . . . , 0) −→ PnC(a0, b0, . . . , an, bn) 7→ [a0 + ib0 : . . . : an + ibn]


with the map ψ is a polynomial map, hence Q-semi-algebraic by Example 2.6.3.Thus

Imψ π = Imψ ⊆ RN

is Q-semi-algebraic by Fact 2.6.4.

Second step (zero set of a polynomial): We use the notation

V (g) := x ∈ PnC | g(x) = 0 for g ∈ C[x0, . . . , xn] homogenous, and

W (h) := t ∈ RN |h(t) = 0 for h ∈ C[y00, . . . , znn].

Let Xan = V (g) for some homogenous g ∈ Q[x0, . . . , xn]. Then ψ(Xan) ⊆ RN is

a Q-semi-algebraic subset, as a little calculation shows. Setting for k = 0, . . . , n

gk := “g(xxk)”

= g(x0xk, . . . , xnxk)

= g((a0ak + b0bk) + i(b0ak − a0bk), . . . , (anak + bnbk) + i(bnak − anbk)

),

where xj = aj + ibj for j = 0, . . . , n, and

hk := g(y0k + iz0k, . . . , ynk + iznk),

we obtain

ψ(Xan) = ψ(V (g))

=

n⋂k=0

ψ(V (gk))

=

n⋂k=0

ψ(PnC) ∩W (hk)

=

n⋂k=0

ψ(PnC) ∩W (Rehk) ∩W (Imhk).

Final step: We can choose an embedding

X ⊆ PnQ,

thus gettingXan ⊆ PnC.

Since X is a locally closed subvariety of PnQ

, the space Xan can be expressed in

terms of subvarieties of the form V (g) with g ∈ Q[x0, . . . , xn], using only thefollowing basic operations

1. complementary set,


2. finite intersection,

3. finite union.

Now Q-semi-algebraic sets are stable under these operations as well and the firstassertion is proved.

Second assertion: The first part of the lemma provides us with Q-semi-algebraicinclusions

ψ : Xan ⊆ PnCx=[x0:...:xn]

⊆ RN(y00,z00,...,ynn,znn)

,

φ : Y an ⊆ PmCu=[u0:...:um]

⊆ RM(v00,w00,...,vmm,wmm)

,

and a choice of coordinates as indicated. We use the notation

V (g) := (x, u) ∈ PnC × PmC | g(x, u) = 0,for g ∈ C[x0, . . . , xn, u0, . . . , um] homogenous in both x and u, and

W (h) := t ∈ RN+M |h(t) = 0, for h ∈ C[y00, . . . , znn, v00, . . . , wmm].

Let Ui be a finite open affine covering of X such that f(Ui) satisfies

• f(Ui) does not meet the hyperplane uj = 0 ⊂ PmQ

for some j, and

• f(Ui) is contained in an open affine subset Vi of Y .

This is always possible, since we can start with the open covering Y ∩ uj 6= 0of Y , take a subordinated open affine covering Vi′, and then choose a finiteopen affine covering Ui subordinated to f−1(Vi′). Now each of the maps

fi := fan|Ui : Uan

i −→ Y an

has image contained in V ani and does not meet the hyperplane u ∈ PmC |uj = 0

for an appropriate jfi : Uan

i −→ V ani .

Being associated to an algebraic map between affine varieties, this map is ratio-nal

fi : x 7→

[g′0(x)

g′′0 (x): . . . :

g′j−1(x)

g′′j−1(x): 1j

:g′j+1(x)

g′′j+1(x): . . . :

g′m(x)

g′′m(x)

],

with g′k, g′′k ∈ Q[x0, . . . , xn], k = 0, . . . , j, . . . ,m. Since the graph Γfan of fan

is the finite union of the graphs Γfi of the fi, it is sufficient to prove that

(ψ × φ)(Γfi) is a Q-semi-algebraic subset of RN+M . Now

Γfi = (Uani × V an

i )∩n⋂k=0k 6=j

V

(ykyj− g′k(x)

g′′k (x)

)= (Uan

i × V ani )∩

n⋂k=0k 6=j

V (ykg′′k (x)−yjg′k(x)),


so all we have to deal with is

V (ykg′′k (x)− yjg′k(x)).

Again a little calculation is necessary. Setting

gpq := “ukuqg′′k (xxp)− ujuqg′k(xxp)”

= ukuqg′′k (x0xp, . . . , xnxp)− ujuqg′k(x0xp, . . . , xnxp)

=((ckcq + dkdq) + i(dkcq − ckdq)

)g′′k((a0ap + b0bp) + i(b0ap − a0bp), . . . , (anap + bnbp) + i(bnap − anbp)

)−((cjcq + djdq) + i(djcq − cjdq)

)g′k((a0ap + b0bp) + i(b0ap − a0bp), . . . , (anap + bnbp) + i(bnap − anbp)

),

where xl = al + ibl for l = 0, . . . , n, ul = cl + idl for l = 0, . . . ,m, and

hpq := (vkq+iwkq)g′′k (y0p+iz0p, . . . , ynp+iznp)−(vjq+iwjq)g

′k(y0p+iz0p, . . . , ynp+iznp),

we obtain

(ψ × φ)(V(ykg′′k (x)−yjg′k(x)

))=

=

n⋂p=0

m⋂q=0

(ψ × φ)(V (gpq))

=

n⋂p=0

m⋂q=0

(ψ × φ)(Uani × V an

j ) ∩W (hpq)

=

n⋂p=0

m⋂q=0

(ψ × φ)(Uani × V an

j ) ∩W (Rehpq) ∩W (Imhpq).

2.6.2 Semi-algebraic singular chains

We need further prerequisites in order to state the promised Proposition 2.6.8.

Definition 2.6.7 ( [Hi2, p. 168]). By an open simplex 4we mean the interiorof a simplex (= the convex hull of r+1 points in Rn which span an r-dimensionalsubspace). For convenience, a point is considered as an open simplex as well.

The notation 4stdd will be reserved for the closed standard simplex spanned by

the standard basis

ei = (0, . . . , 0, 1i, 0, . . . , 0) | i = 1, . . . , d+ 1

of Rd+1.


Consider the following data (∗):

• X a variety defined over Q,

• D a divisor in X with normal crossings,

• and finally γ ∈ Hsingp (Xan, Dan;Q), p ∈ N0.

As before, we have denoted by Xan (resp. Dan) the complex analytic spaceassociated to the base change XC = X ×Q C (resp. DC = D ×Q C).

By Lemma 2.6.5, we may consider both Xan and Dan as bounded Q-semi-algebraic subsets of RN .

We are now able to formulate our proposition.

Proposition 2.6.8. With data (∗) as above, we can find a representative ofγ that is a rational linear combination of singular simplices each of which isQ-semi-algebraic.

The proof of this proposition relies on the following proposition due to Lo-jasiewicz which has been written down by Hironaka.

Proposition 2.6.9 ( [Hi2, p. 170]). For Xi a finite system of bounded Q-semi-algebraic sets in Rn, there exists a simplicial decomposition

Rn =∐j

4j

by open simplices 4j and a Q-semi-algebraic automorphism

κ : Rn → Rn

such that each Xi is a finite union of some of the κ(4j).

Note 2.6.10. Although Hironaka considers R-semi-algebraic sets, we can safelyreplace R by Q in this result (including the fact he cites from [Sb]). The onlyproblem that could possibly arise concerns a “good direction lemma”:

Lemma 2.6.11 (Good direction lemma for R, [Hi2, p. 172], [KB, Thm. 5.I,p. 242]). Let Z be a R-semi-algebraic subset of Rn, which is nowhere dense. Adirection v ∈ Pn−1

R (R) is called good, if any line l in Rn parallel to v meets Zin a discrete (maybe empty) set of points; otherwise v is called bad. Then theset B(Z) of bad directions is a Baire category set in Pn−1

R (R).

This gives immediately good directions v ∈ Pn−1R (R)\B(Z), but not necessarily

v ∈ Pn−1

Q(Q) \ B(Z). However, in Remark 2.1 of [Hi2], which follows directly

after the lemma, the following statement is made: If Z is compact, then B(Z)

is closed in Pn−1R (R). In particular Pn−1

Q(Q)\B(Z) will be non-empty. Since we


only consider bounded Q-semi-algebraic sets Z ′, we may take Z := Z ′ (which is

compact by Heine-Borel), and thus find a good direction v ∈ Pn−1

Q(Q) \ B(Z ′)

using B(Z ′) ⊆ B(Z). Hence:

Lemma 2.6.12 (Good direction lemma for Q). Let Z ′ be a bounded Q-semi-algebraic subset of Rn, which is nowhere dense. Then the set Pn−1

R (R) \ B(Z)of good directions is non-empty.

Proof of Proposition 2.6.8. Applying Proposition 2.6.9 to the two-element sys-tem of Q-semi-algebraic sets Xan, Dan ⊆ RN , we obtain a Q-semi-algebraicdecomposition

RN =∐j

4j

of RN by open simplices 4j and a Q-semi-algebraic automorphism

κ : RN → RN .

We write 4j for the closure of 4j . The sets

K := 4j |κ(4j) ⊆ Xan and L := 4j |κ(4j) ⊆ Dan

can be thought of as finite simplicial complexes, but built out of open simplicesinstead of closed ones. We define their geometric realizations

|K| :=⋃4j∈K

4j and |L| :=⋃4j∈L

4j .

Then Proposition 2.6.9 states that κmaps the pair of topological spaces (|K|, |L|)homeomorphically to (Xan, Dan).

Easy case: If X is complete, so is XC (by [Ha2, Cor. II.4.8(c), p. 102]), henceXan and Dan will be compact [Ha2, B.1, p. 439]. In this situation,

K := 4j |κ(4j) ⊆ Xan and L := 4j |κ(4j) ⊆ Dan

are (ordinary) simplicial complexes, whose geometric realizations coincide withthose of K and L, respectively. In particular

Hsimpl• (K,L;Q) ∼= Hsing

• (∣∣K∣∣ , ∣∣L∣∣ ;Q)

∼= Hsing• (|K|, |L|;Q)

∼= Hsing• (Xan, Dan;Q).

(2.4)

Here Hsimpl• (K,L;Q) denotes simplicial homology of course.

We write γsimpl ∈ Hsimplp (K,L;Q) and γsing ∈ Hsing

p (∣∣K∣∣ , ∣∣L∣∣ ;Q) for the image

of γ under this isomorphism. Any representative Γsimpl of γsimpl is a rationallinear combination

Γsimpl =∑j aj4j , aj ∈ Q


of oriented closed simplices 4j ∈ K. We can choose orientation-preservingaffine-linear maps of the standard simplex 4std

p to 4j

σj : 4stdp −→ 4j for 4j ∈ Γsimpl.

These maps yield a representative

Γsing :=∑j aj σj

of γsing. Composing with κ yields Γ := κ∗Γsing ∈ γ, where Γ has the desiredproperties.

In the general case, we perform a barycentric subdivision B on K twice (onceis not enough) and define |K| and |L| not as the “closure” of K and L, but asfollows (see Figure 2.1)

K := 4 |4∈ B2(K) and 4 ⊆ |K|,L := 4 |4∈ B2(K) and 4 ⊆ |L|.

(2.5)

κ−1(Xan) ∩4j K ∩4j∣∣K∣∣ ∩4j

Intersection of κ−1(Xan) witha closed 2-simplex 4j , wherewe assume that part of theboundary ∂4j does not be-long to κ−1(Xan)

Open simplices of K con-tained in 4j

Intersection of∣∣K∣∣ with 4j

(the dashed lines show thebarycentric subdivision)

Figure 2.1: Definition of K

The point is that the pair of topological spaces (∣∣K∣∣ , ∣∣L∣∣) is a strong deformation

retract of (|K|, |L|). Assuming this, we see that in the general case with K, Ldefined as in (2.5), the isomorphism (2.4) still holds and we can proceed as inthe easy case to prove the proposition.

We define the retraction map

ρ : (|K| × [0, 1], |L| × [0, 1])→ (∣∣K∣∣ , ∣∣L∣∣)

as follows: Let 4j ∈ K be an open simplex which is not contained in theboundary of any other simplex of K and set

inner := 4j ∩K, outer := 4j \K.


Figure 2.2: Definition of qp

Note that inner is closed. For any point p ∈ outer the ray −→c p from the centerc of 4j through p “leaves” the set inner at a point qp, i.e. −→c p ∩ inner equalsthe line segment c qp; see Figure 2.2. The map

ρj : 4j × [0, 1]→4j

(p, t) 7→

p if p ∈ inner,qp + t · (p− qp) if p ∈ outer

retracts 4j onto inner.

Now these maps ρj glue together to give the desired homotopy ρ.

We want to state one of the intermediate results of this proof explicitly:

Corollary 2.6.13. Let X and D be as above. Then the pair of topologicalspaces (Xan, Dan) is homotopy equivalent to a pair of (realizations of) simplicialcomplexes (|Xsimpl|, |Dsimpl|).

2.7 Singular cohomology via the h′-topology

In order to give a simple description of the period isomorphism for singularvarieties, we are going to need a more sophisticated description of singularcohomology.

We work in the category of complex analytic spaces An.

Definition 2.7.1. Let X be a complex analytic space. The h′-topology on thecategory (An/X)h′ of complex analytic spaces overX is the smallest Grothendiecktopology such that the following are covering maps:

1. proper surjective morphisms;

2. open covers.

2.7. SINGULAR COHOMOLOGY VIA THE H′-TOPOLOGY 85

If F is a presheaf of An/X we denote Fh′ its sheafification in the h′-topology.

Remark 2.7.2. This definition is inspired by Voevodsky’s h-topology on thecategory of schemes, see Section 3.2. We are not sure if it is the correct analoguein the analytic setting. However, it is good enough for our purposes.

Lemma 2.7.3. For Y ∈ An let CY be the (ordinary) sheaf associated to thepresheaf C. Then

Y 7→ CY (Y )

is an h′-sheaf on An.

Proof. We have to check the sheaf condition for the generators of the topology.By assumption it is satisfied for open covers. Let Y → Y be proper surjective.Without loss of generality Y is connected. Let Yi for i ∈ I be the collection ofconnected components of Y . Then

Y ×Y Y =⋃i,j∈I

Yi ×Y Yj

We have to compute the kernel of∏i∈I

C→∏i,j

C(Yi ×Y Yj)

via the difference of the two natural restriction maps. Comparing ai and ajin C(Yi ×Y Yj) we see that they agree. Hence the kernel is just one copy ofC = CY (Y ).

Proposition 2.7.4. Let X be an analytic space and i : Z ⊂ X a closed sub-space. Then there is a morphism of sites ρ : (An/X)h′ → X. It induces anisomorphism

Hising(X,Z;C)→ Hi

h′((An/X)h′ ,Ker(Ch′ → i∗Ch′))

compatible with long exact sequences and products.

Remark 2.7.5. This statement and the following proof can be extended tomore general sheaves F .

The argument uses the notion of a hypercover, see Definition 1.5.8.

Proof. We first treat the absolute case with Z = ∅. We use the theory of co-homological descent as developed in [SGA4Vbis]. Singular cohomology satisfiescohomological descent for open covers and also for proper surjective maps (seeTheorem 2.7.6). (The main ingredient for the second case is the proper basechange theorem.) Hence it satisfies cohomological descent for h′-covers. Thisimplies that singular cohomology can be computed as a direct limit

limX•

C(X•),


where X• runs through all h′-hypercovers. On the other hand, the same limitcomputes h′-cohomology, see Proposition 1.6.9 For the general case, recall thatwe have a short exact sequence

0→ j!C→ C→ i∗C→ 0

of sheaves on X. Its pull-back to An/X maps naturally to the short exactsequence

0→ Ker(Ch′ → i∗Ch′))→ Ch′ → i∗Ch′ → 0 .

This reduces the comparison in the relative case to the absolute case oncewe have shown that Ri∗Ch′ = i∗Ch′ . The sheaf Rni∗Ch′ is given by the h′-sheafification of the presheaf

X ′ 7→ Hnh′(Z ×X X ′,Ch′) = Hn

sing(Z ×X X ′,C)

for X ′ → X in An/X. By resolution of singularities for analytic spaces wemay assume that X ′ is smooth and Z ′ = X ′ ×X Z a divisor with normal cross-ings. By passing to an open cover, we may assume that Z ′ an open ball in aunion of coordinate hyperplanes, in particular contractible. Hence its singularcohomology is trivial. This implies that Rni∗Ch′ = 0 for n ≥ 1.

Theorem 2.7.6 (Descent for proper hypercoverings). Let D ⊂ X be a closedsubvariety and D• → D a proper hypercover(see Definition 1.5.8), such thatthere is a commutative diagram

D• −−−−→ X•y yD −−−−→ X

Then one has cohomological descent for singular cohomology:

H∗(X,D;Z) = H∗ (Cone(Tot(X•)→ Tot(D•))[−1];Z) .

Here, Tot(−) denotes the total complex in Z[Var] associated to the correspondingsimplicial variety, see Definition 1.5.11.

Proof. The relative case follows from the absolute case. The essential ingredientis proper base change, which allows to reduce to the case where X is a point. Thestatement then becomes a completely combinatorial assertion on contractibilityof simplicial sets. The results are summed up in [D5] (5.3.5). For a completereference see [SGA4Vbis], in particular Corollaire 4.1.6.

Chapter 3

Algebraic de Rhamcohomology

Let k be a field of characteristic zero. We are going to define relative algebraicde Rham cohomology for general varieties over k, not necessarily smooth.

3.1 The smooth case

In this section, all varieties are smooth over k. In this case, de Rham cohomologyis defined as hypercohomology of the complex of sheaves of differentials.

3.1.1 Definition

Definition 3.1.1. Let X be a smooth variety over k. Let Ω1X be the sheaf of

k-differentials on X. For p ≥ 0 let

ΩpX = ΛpΩ1X

be the exterior power in the category ofOX -modules. The universal k-derivationd : OX → Ω1

X induces

dp : ΩpX → Ωp+1X .

We call (Ω•X , d) the algebraic de Rham complex of X.

If X is smooth of dimension n, the sheaf Ω1X is locally free of rank n. This allows

to define exterior powers. Note that ΩiX vanishes for i > n. The differential isuniquely characterized by the properties:

1. d0 = d on OX ;

87

88 CHAPTER 3. ALGEBRAIC DE RHAM COHOMOLOGY

2. dp+1dp = 0 for all p ≥ 0;

3. d(ω ∧ ω′) = dω ∧ ω′ + (−1)pω ∧ dω′ for all local sections ω of ΩpX and ω′

of Ωp′

X .

Indeed, if t1, . . . , tn is a system of local parameters at x ∈ X, then local sectionsof ΩpX near x can be expressed as

ω =∑

1≤i1<···<ip≤n

fi1...ipdti1 ∧ · · · ∧ dtip

and we have

dpω =∑

1≤i1<···<ip≤n

dfi1...ip ∧ dti1 ∧ · · · ∧ dtip .

Definition 3.1.2. Let X be smooth variety over a field k of characteristic 0.We define algebraic de Rham cohomology of X as the hypercohomology

HidR(X) = Hi(X,Ω•X) .

For background material on hypercohomology see Section 1.4.

If X is smooth and affine, this simplifies to

HidR(X) = Hi(Ω•X(X)) .

Example 3.1.3. 1. Consider the affine line X = A1k = Spec k[t]. Then

Ω•A1(A1) =[k[t]

d−→ k[t]dt].

We have

Ker(d) = P ∈ k[t]|P ′ = 0 = k , Im(d) = k[t]dt ,

because we have assumed characteristic zero. Hence

HidR(A1) =

k i = 0,

0 i > 0.

2. Consider the multiplicative group X = Gm = Spec k[t, t−1]. Then

Ω•Gm(Gm) =[k[t, t−1]

d−→ k[t, t−1]dt].

We have

Ker(d) = P ∈ k[t]|P ′ = 0 = k ,

Im(d) = N∑i=n

aitidt|a−1 = 0 ,

3.1. THE SMOOTH CASE 89

again because of characteristic zero. Hence

HidR(Gm) =

k i = 0, 1,

0 i > 1.

The isomorphism for i = 1 is induced by the residue for meromorphicdifferential forms.

3. Let X be a connected smooth projective curve of genus g. We use thestupid filtration on the de Rham complex

0→ Ω1X [−1]→ Ω•X → OX [0]→ 0 .

The cohomological dimension of any variety X is the index i above whichthe cohomology Hi(X,F) of any coherent sheaf F vanishes, see [Ha2,Chap. III, Section 4]. The cohomological dimension of a smooth, projec-tive curve is 1, hence the long exact sequence reads

0→ H−1(X,Ω1X)→ H0

dR(X)→ H0(X,OX)

∂−→ H0(X,Ω1X)→ H1

dR(X)→ H1(X,OX)

∂−→ H1(X,Ω1X)→ H2

dR(X)→ 0

This is a special case of the Hodge spectral sequence. It is known todegenerate (e.g. [D4]). Hence the boundary maps ∂ vanish. By Serreduality, this yields

HidR(X) =

H0(X,OX) = k i = 0,

H1(X,Ω1X) = H0(X,OX)∨ = k i = 2,

0 i > 2.

The most interesting group for i = 1 sits in an exact sequence

0→ H0(X,Ω1X)∨ → H1

dR(X)→ H0(X,Ω1X)→ 0

and hence

dimH1dR(X) = 2g .

Remark 3.1.4. In these cases, the explicit computation shows that algebraicde Rham cohomology computes the standard Betti numbers of these varieties.We are going to show in chapter 5 that this is always true. In particular, itis always finite dimensional. A second algebraic proof of this fact will also begiven in Corollary 3.1.17.

Lemma 3.1.5. Let X be a smooth variety of dimension d. Then HidR(X)

vanishes for i > 2d. If in addition X is affine, it vanishes for i > d.


Proof. We use the stupid filtration on the de Rham complex. This induces asystem of long exact sequences relating the groups Hi(X,ΩpX) to algebraic deRham cohomology.

Any variety of dimension d has cohomological dimension ≤ d for coherentsheaves [Ha2, ibid.]. All ΩpX are coherent, hence Hi(X,ΩpX) vanishes for i > d.The complex Ω•X is concentrated in degrees at most d. This adds up to cohomo-logical dimension 2d for algebraic de Rham cohomology. Affine varieties havecohomological dimension 0, hence Hi(X,ΩpX) vanishes already for i > 0.

3.1.2 Functoriality

Let f : X → Y be morphism of smooth varieties over k. We want to explainthe functoriality

f∗ : HidR(Y )→ Hi

dR(X) .

We use the Godement resolution (see Definition 1.4.8) and put

RΓdR(X) = Γ(X,Gd(Ω•X)) .

The natural map f−1OX → OX induces a unique multiplicative map

f−1Ω•X → Ω•Y .

By functoriality of the Godement resolution, we have

f−1GdX(Ω•X)→ GdY (f−1Ω•X)→ GdY (Ω•Y ) .

Taking global sections, this yields

RΓdR(Y )→ RΓdR(X) .

We have shown:

Lemma 3.1.6. De Rham cohomology HidR(·) is a contravariant functor on the

category of smooth varieties over k with values in k-vector spaces. It is inducedby a functor

RΓdR : Sm→ C+(k−Mod) .

Note that Q ⊂ k, so the functor can be extended Q-linearly to Q[Sm]. Thisallows to extend the definition of algebraic de Rham cohomology to complexes ofsmooth varieties in the next step. Explicitly: let X• be an object of C−(Q[Sm]).Then there is a double complex K•,• with

Kn,m = Γ(X−n, Gdm(Ω•)) .

Definition 3.1.7. Let X• be a object of C−(Z[Sm]). We denote the totalcomplex by

RΓdR(X•) = Tot(K•,•)

and setHi

dR(X•) = Hi(RΓdR(X•) .

We call this the algebraic de Rham cohomology of X•.


3.1.3 Cup product

Let X be a smooth variety over k. Wedge product of differential forms turnsΩ•X into a differential graded algebra:

Tot(Ω•X ⊗k Ω•X)→ Ω•X .

The compatibility with differentials was built into the definition of d in Defini-tion 3.1.1.

Lemma 3.1.8. H•dR(X) carries a natural multiplication

∪ : HidR(X)⊗k Hj

dR(X)→ Hi+jdR (X)

induced from wedge product of differential forms.

Proof. We need to define

RΓdR(X)⊗k RΓdR(X)→ RΓdR(X)

as a morphism in the derived category. We have quasi-isomorphisms

Ω•X ⊗ Ω•X → Gd(Ω•X)⊗Gd(Ω•X)

and hence a quasi-isomorphism of flasque resolutions of Ω•X ⊗ Ω•X

s : Gd(Ω•X ⊗ Ω•X)→ Gd (Gd(Ω•X)⊗Gd(Ω•X))

In the derived category, this allows the composition

RΓdR(X)⊗k RΓdR(X) = Γ(X,Gd(Ω•X))⊗k Γ(X,Gd(Ω•X))

→ Γ(X,Gd(Ω•)⊗Gd(Ω•X))

→ Γ (X,Gd (Gd(Ω•X)⊗Gd(Ω•X)))

← sΓ(X,Gd(Ω•X ⊗ Ω•X))

→ Γ(X,Gd(Ω•X)) = RΓdR(X) .

The same method also allows the construction of an exterior product.

Proposition 3.1.9 (Kunneth formula). Let X,Y be smooth varieties. There isa natural multiplication induced from wedge product of differential forms

HidR(X)⊗k Hj

dR(Y )→ Hi+jdR (X × Y ) .

It induces an isomorphism

HndR(X × Y ) ∼=

⊕i+j=n

HidR(X)⊗k Hj

dR(Y ) .


Proof. Let p : X × Y → X and q : X × Y → Y be the projection maps. Theexterior multiplication is given by

HidR(X)⊗Hj

dR(Y )p∗⊗q∗−−−−→ Hi

dR(X × Y )⊗HjdR(X × Y )

∪−→ Hi+jdR (X × Y ) .

The Kunneth formula is most easily proved by comparison with singular coho-mology. We postpone the proof to Lemma 5.3.2 in Chap. 5.

Corollary 3.1.10 (Homotopy invariance). Let X be a smooth variety. Thenthe natural map

HndR(X)→ Hn

dR(X × A1)

is an isomorphism.

Proof. We combine the Kunneth formula with the compuation in the case of A1

in Example 3.1.3.

3.1.4 Change of base field

Let K/k be an extension of fields of characteristic zero. We have the corre-sponding base change functor

X 7→ XK

from (smooth) varieties over k to (smooth) varieties over K. Let

π : XK → X

be the natural map of schemes. By standard calculus of differential forms,

Ω•XK/K∼= π∗Ω•X/k = π−1Ω•X/k ⊗k K .

Lemma 3.1.11. Let K/k be an extension of fields of characteristic zero. LetX be a smooth variety over k. Then there are natural isomorphisms

HidR(X)⊗k K → Hi

dR(XK) .

They are induced by a natural quasi-isomorphism

RΓdR(X)⊗k K → RΓdR(XK) .

Proof. By functoriality of the Godement resolution (see Lemma 1.4.10) andk-linarity, we get natural quasi-isomorphisms

π−1GdX(Ω•X/k)⊗k K → GdXK (π−1Ω•X/k)→ GdXK (Ω•XK/K) .


AsK is flat over k, taking global sections induces a sequence of quasi-isomorphisms

RΓdR(X)⊗k K = Γ(X,GdX(Ω•X/k))⊗k K∼= Γ(XK , π

−1GdX(Ω•X/k)⊗k K∼= Γ(XK , π

−1GdX(Ω•X/k)⊗k K)

→ Γ(XK , GdXK (Ω•XK/K)

= RΓdR(XK) .

Remark 3.1.12. This immediately extends to algebraic de Rham cohomologyof complexes of smooth varieties.

Conversely, we can also restrict scalars.

Lemma 3.1.13. Let K/k be a finite field extension. Let Y be a smooth varietyover K. Then there are a natural isomorphism

HidR(Y/k)→ Hi

dR(Y/K).

They are induced by a natural isomorphism

RΓdR(Y/k)→ RΓdR(Y/K).

Proof. We use the sequence of sheaves on Y ([Ha2] Proposition 8.11)

π∗Ω1K/k → Ω1

Y/k → Ω1Y/K → 0

where π : Y → SpecK is the structural map. As we are in characteristic 0, wehave Ω1

K/k = 0. This implies that we actually have identical de Rham complexes

Ω•Y/K = Ω•Y/k

and identical Godement resolutions.

3.1.5 Etale topology

In this section, we give an alternative interpretation of algebraic de Rham co-homology using the etale topology. The results are not used in our discussionsof periods.

Let Xet be the small etale site on X, see section 1.6. The complex of differentialforms Ω•X can be viewed as a complex of sheaves on Xet (see [Mi], Chap. II,Example 1.2 and Proposition 1.3). We write Ω•Xet

for distinction.

Lemma 3.1.14. There is a natural isomorphism

HidR(X)→ Hi(Xet,Ω

•Xet

) .


Proof. The map of sites s : Xet → X induces a map on cohomology

Hi(X,Ω•X)→ Hi(Xet,Ω•Xet

) .

We filter Ω•X by the stupid filtration F pΩ•X

0→ F p+1Ω•X → F pΩ•X → ΩpX [−p]→ 0

and compare the induced long exact sequences in cohomology on X and Xet.As the ΩpX are coherent, the comparison maps

Hi(X,ΩpX)→ Hi(Xet,ΩpXet

)

are isomorphisms by [Mi] Chap. III, Proposition 3.7. By descending inductionon p, this implies that we have isomorphisms for all F pΩ•X , in particular for Ω•Xitself.

3.1.6 Differentials with log poles

We give an alternative description of algebraic de Rham cohomology using dif-ferentials with log poles as introduced by Deligne, see [D4], Chap. 3. We arenot going to use this point of view in our study of periods.

Let X be a smooth variety and j : X → X an open immersion into a smoothprojective variety such that D = XrX is a simple divisor with normal crossings(see Definition 1.1.2).

Definition 3.1.15. Let

Ω1X〈D〉 ⊂ j∗Ω

1X

be the locally free OX -module with the following basis: if U ⊂ X is an affineopen subvariety etale over An via coordinates t1, . . . , tn and D|U given by theequation t1 . . . tr = 0, then Ω1

X〈D〉|U has OX -basis

dt1t1, . . . ,

dtrtr, dtr+1, . . . , dtn .

For p > 1 let

ΩpX〈D〉 = ΛpΩ1

X〈D〉 .

We call the Ω•X〈D〉 the complex of differentials with log poles along D.

Note that the differential of j∗Ω•X respects Ω•

X〈D〉, so that this is indeed a

complex.

Proposition 3.1.16. The inclusion induces a natural isomorphism

Hi(X,Ω•X〈D〉)→ Hi(X,Ω•X) .


Proof. This is the algebraic version of [D4], Prop. 3.1.8. We indicate the argu-ment. Note that j : X → X is affine, hence j∗ is exact and we have

Hi(X,Ω•X) ∼= Hi(X, j∗Ω•X) .

It remains to show that

ι : Ω•X〈D〉 → j∗Ω•X

is a quasi-isomorphism, or, equivalently, that Coker(ι) is exact. By Lemma3.1.14 we can work in the etale topology. It suffices to check exactness in stalksin geometric points of X over closed points. As X is smooth and D a divisorwith normal crossings, it suffices to consider the case D = V (t1 . . . tr) ⊂ An andthe stalk in 0. As in the proof of the Poincare lemma, it suffices to consider thecase n = 1. If r = 0, then there is nothing to show.

In remains to consider the following situation: let k = k, O be the henselizationof k[t] with respect to the ideal (t). We have to check that the complex

O[t−1]/O → O[t−1]/t−1Odt

is acyclic. The term in degree 0 has the O-basis t−i for i > 0. The term indegree 1 has the O-basis t−idt for i > 1. In this basis, the differential has theform

fdt

ti7→

f ′ dtti − if

dtti+1 i > 1,

−f dtt2 i = 1.

It is injective because char(k) = 0. By induction on i we also check that it issurjective.

Corollary 3.1.17. Let X be a smooth variety over k. Then the algebraic deRham cohomology groups Hi

dR(X) are finite dimensional k-vector spaces.

Proof. By resolution of singularities, we can embed X into a projective X suchthat D is a simple divisor with normal crossings. By the proposition

HidR(X) = Hi(X,Ω•X〈D〉) .

Note that all Ω•X〈D〉 are coherent sheaves on a projective variety, hence the

cohomology groups Hp(X,ΩqX〈D〉) are finite dimensional over k. We use the

stupid filtration on Ω•X〈D〉 and the induced long exact cohomology sequence.

By induction, all Hq(X, F pΩ•X〈D〉) are finite dimensional.

Remark 3.1.18. The complex of differentials with log poles is studied inten-sively in the theory of mixed Hodge structures. Indeed, Deligne uses it in [D4] inorder to define the Hodge and the weight filtration on cohomology of a smoothvariety X. We are not going to use Hodge structures in the sequel though.


3.2 The general case: via the h-topology

We now want to extend the definition to the case of singular varieties and evento relative cohomology. The most simple minded idea – use Definition 3.1.2 –does not give the desired dimensions.

Example 3.2.1. Consider X = SpecA with A = k[X,Y ]/XY , the union oftwo affine lines. This variety is homotopy equivalent to a point, so we expect itscohomology to be trivial. We compute the cohomology of the de Rham complex

A→ 〈dX, dY 〉A/〈XdY + Y dX〉A .

Elements of A can be represented uniquely by polynomials of the form

P =

n∑i=0

aiXi +

m∑j=1

bjYj

with

dP =

n∑i=1

iaiXi−1dX +

m∑j=1

bjjYj−1dY .

P is in the kernel of d if it is constant. On the other hand d is not surjectivebecause it misses differentials of the form Y idX.

There are different ways of adapting the definition in order to get a well-behavedtheory.

The h-topology introduced by Voevodsky makes the handling of singular vari-eties straightforward. In this topology, any variety is locally smooth by resolu-tion of singularities. The h-sheafification of the presheaf of Kahler differentialswas studied in detail by Huber and Jorder in [HJ]. The weaker notion of eh-differential was already introduced by Geisser in [Ge].

We review a definition given by Voevodsky in [Voe].

Definition 3.2.2 ([Voe] Section 3.1). A morphism of schemes p : X → Yis called topological epimorphism if Y has the quotient topology of X. It isa universal topological epimorphism if any base change of p is a topologicalepimorphism.

The h-topology on the category (Sch/X)h of separated schemes of finite typeover X is the Grothendieck topology with coverings finite families pi : Ui → Y such that

⋃i Ui → Y is a universal topological epimorphism.

By [Voe] the following are h-covers:

1. finite flat covers (in particular etale covers);


3.2. THE GENERAL CASE: VIA THE H-TOPOLOGY 97

3. quotients by finite groups actions.

The assignment X 7→ ΩpX/k(X) is a presheaf on Sch. We denote by Ωph (resp.

Ωph/X , if X needs to be specified) its sheafification in the h-topology, and by

Ωph(X) its value as abelian group.

Definition 3.2.3. Let X be a separated k-scheme of finite type over k. Wedefine

HidR(Xh) = Hi((Sch/X)h,Ω

•h) .

Proposition 3.2.4 ([HJ] Theorem 3.6, Proposition 7.4). Let X be smooth overk. Then

Ωph(X) = ΩpX/k(X)

andHi

dR(Xh) = HidR(X) .

Proof. The statement on Ωph(X) is [HJ], Theorem 3.6. The statement on the deRham cohomology is loc.cit., Proposition 7.4. together with the comparison ofloc. cit., Lemma 7.22.

Remark 3.2.5. The main ingredients of the proof are a normal form for h-covers established by Voevodsky in [Voe] Theorem 3.1.9, an explicit computationfor the blow-up of a smooth variety in a smooth center and etale descent for thecoherent sheaves ΩpY/k.

A particular useful h-cover are abstract blow-ups, covers of the form (f : X ′ →X, i : Z → X) where Z is a closed immersion and f is proper and an isomor-phism above X − Z.

Then, the above implies that there is a long exact blow-up sequence

. . .→ HidR(X)→ Hi

dR(X ′)⊕HidR(Z)→ Hi

dR(f−1(Z))→ . . .

induced by the blow-up triangle

[f−1(Z)]→ [X ′]⊕ [Z]→ [X]

in SmCor.

Definition 3.2.6. Let X ∈ Sch and i : Z → X a closed subscheme. Put

Ωph/(X,Z) = Ker(Ωph/X → i∗Ωph/Z)

in the category of abelian sheaves on (Sch/X)h.

We define relative algebraic de Rham cohomology as

HpdR(X,Z) = Hp

h(X,Ω•h/(X,Z)) .


Lemma 3.2.7 ([HJ] Lemma 7.26). Let i : Z → X be a closed immersion.

1. ThenRi∗Ω

ph/Z = i∗Ω

ph/Z

and henceHq

h(X, i∗Ωph/Z) = Hq

h(Z,Ωph) .

2. The natural map of sheaves of abelian groups on (Sch/X)h

Ωph/X → i∗Ωph/Z

is surjective.

Remark 3.2.8. The main ingredient of the proof is resolution of singularitiesand the computation of Ωph(Z) for Z a divisor with normal crossings: it is givenas Kahler differentials modulo torsion, see [HJ] Proposition 4.9.

Proposition 3.2.9 ((Long exact sequence) [HJ] Proposition 2.7). Let Z ⊂ Y ⊂X be closed immersions. Then there is a natural long exact sequence

· · · → HqdR(X,Y )→ Hq

dR(X,Z)→ HqdR(Y,Z)→ Hq+1

dR (X,Y )→ · · ·

Remark 3.2.10. The sequence is the long exact cohomology sequence attachedto the exact sequence of h-sheaves on X

0→ Ωph/(X,Y ) → Ωph/(X,Z) → iY ∗Ωph/(Y,Z) → 0

where iY : Y → X is the closed immersion.

Proposition 3.2.11 ((Excision) [HJ] Proposition 7.28). Let π : X → X be aproper surjective morphism, which is an isomorphism outside of Z ⊂ X. LetZ = π−1(Z). Then

HqdR(X, Z) ∼= Hq

dR(X,Z) .

Remark 3.2.12. This is an immediate consequence of the blow-up triangle.

Proposition 3.2.13 ((Kunneth formula) [HJ] Proposition 7.29). Let Z ⊂ Xand Z ′ ⊂ X ′ be closed immersions. Then there is a natural isomorphism

HndR(X ×X ′, X × Z ′ ∪ Z ×X ′) =

⊕a+b=n

HadR(X,Z)⊗k Hb

dR(X ′, Z ′) .

Proof. We explain the construction of the map. We work in the category ofh-sheaves of k-vector spaces on X ×X ′. Note that h-cohomology of an h-sheafof k-vector spaces computed in the category of sheaves of abelian groups agreeswith its h-cohomology computed in the category of sheaves of k-vector spacesbecause an injective sheaf of k-vector spaces is also injective as sheaf of abeliangroups.

3.2. THE GENERAL CASE: VIA THE H-TOPOLOGY 99

We abbreviate T = X × Z ′ ∪ Z × X ′. By h-sheafification of the product ofKahler differentials we have a natural multiplication

pr∗XΩah/X ⊗k pr∗X′Ωbh/X′ → Ωa+b

h/X×X′ .

It induces, with iZ : Z → X, iZ′ : Z ′ → X ′, and i : T → X ×X ′

pr∗XKer(Ωah/X → iZ∗Ωah/Z)⊗kpr∗X′Ker(Ωbh/X′ → iZ′∗Ω

bh/Z′)→ Ker(Ωa+b

h/X×X′ → i∗Ωa+bh/T ) .

The resulting morphism

pr•XΩ∗h/(X,Z) ⊗k pr•X′Ω∗h/(X′,Z′) → Ω•h/(X×X′,T ) .

induces a natural Kunneth morphism⊕a+b=n

HadR(X,Z)⊗k Hb

dR(X ′, Z ′)→ HndR(X ×X ′, T ) .

We refer to the proof of [HJ] Proposition 7.29 for the argument that this is anisomorphism.

Lemma 3.2.14. Let K/k be an extension of fields of characteristic zero. LetX be a variety over k and Z ⊂ X a subvariety. Then there are natural isomor-phisms

HidR(X,Z)⊗k K → Hi

dR(XK , ZK) .


RΓdR(X)⊗k K → RΓdR(XK) .

Proof. Via the long exact cohomology sequence for pairs, and the long exactsequence for a blow-up, it suffices to consider the case when X is a single smoothvariety, where it follows from Lemma 3.1.11.

Lemma 3.2.15. Let K/k be a finite extension of fields of characteristic 0. LetY be variety over K and W ⊂ Y a subvariety. We denote Yk and Wk the samevarieties when considered over k.

Then there are natural isomorphisms

HidR(Y,W )→ Hi

dR(Yk,Wk) .


RΓdR(Y )→ RΓdR(YK) .

Proof. Note that if a variety is smooth over K, then it is also smooth whenviewed over k.

The morphism on cohomology is induced by a morphism of sites from the cate-gory of k-varieties over Y to the category of K-varieties over k, both equippedwith the h-topology. The pull-back of the de Rham complex over Y maps tothe de Rham complex over Yk. Via the long exact sequence for pairs and theblow-up sequence, it suffices to show the isomorphism for a single smooth Y .This was settled in Lemma 3.1.13.


3.3 The general case: alternative approaches

We are now going to present a number of earlier definitions in the literature.They all give the same results in the cases where they are defined.

3.3.1 Deligne’s method

We present the approach of Deligne in [D5]. A singular variety is replaced by asuitable simplicial variety whose terms are smooth.

3.3.2 Hypercovers

See Section 1.5 for basics on simplicial objects. In particular, we have the notionof an S-hypercover for a class of covering maps of varieties.

We will need two cases:

1. S is the class of open covers, i.e., X =∐ni=1 Ui with Ui ⊂ Y open and

such that⋃ni=1 Ui = Y .

2. S the class of proper surjective maps.

Lemma 3.3.1. Let X → Y be in S. We put

X• = cosqY0 X .

In explicit terms,

Xp = X ×Y · · · ×Y X (p+ 1 factors)

where we number the factors from 0 to p. The face map ∂i is the projectionforgetting the factor number i. The degeneration si is induced by the diagonalfrom the factor i into the factors i and i+ 1.

Then X• → Y is an S-hypercover.

Proof. By [SGA4.2] Expose V, Proposition 7.1.2, the morphism

cosq0 → cosqn−1sqn−1cosq0

is an isomorphism of functors for n ≥ 1. (This follows directly from the ad-junction properties of the coskeleton functor.) Hence the condition on Xn issatisfied trivially for n ≥ 1. In degree 0 we consider

X0 = X → (cosqY−1sq−1cosqY0 )0 = Y .

By assumption, it is in S.

3.3. THE GENERAL CASE: ALTERNATIVE APPROACHES 101

It is worth spelling this out in complete detail.

Example 3.3.2. Let X =∐ni=1 Ui with Ui ⊂ Y open. For i0, . . . , ip ∈

1, . . . , n we abbreviate

Ui0,...,ip = Ui0 ∩ · · · ∩ Uip .

Then the open hypercover X• is nothing but

Xp =∐

i0,...,ip=0n

Ui0,...,ip

with face and degeneracy maps given by the natural inclusions. Let F be asheaf of abelian groups on X. Then the complex associated to the cosimplicialabelian group F(X•) is given by

n⊕i=1

F(Ui)→n⊕

i0,i1=1

F(Ui0,i1)→n⊕

i0,i1,i2=1

F(Ui0,i1,i2)→ . . .

with differential

δp(α)i0,...,ip =

p+1∑j=0

αi0,...,ij ,...,ip+1|Ui0,...,ip+1

,

i.e., the differential of the Cech complex. Indeed, the natural projection

F(X•)→ C•(U,F)

to the Cech complex (see Definition 1.4.12) is a quasi-isomorphism.

Definition 3.3.3. We say that X• → Y• is a smooth proper hypercover if it isa proper hypercover with all Xn smooth.

Example 3.3.4. Let Y = Y1 ∪ . . . Yn with Yi ⊂ Y closed. For i0, . . . , ip =1, . . . , n put

Yi0,...,ip = Yi0 ∩ . . . Yip .

Assume that all Yi and all Yi0,...,ip are smooth.

Then X =∐ni=1 Yi → Y is proper and surjective. The proper hypercover X• is

nothing but

Xn =∐

i0,...,in=0n

Yi0 ∩ . . . Yin

with face and degeneracy maps given by the natural inclusions. Hence X• → Yis a smooth proper hypercover. As in the open case, the projection to Cechcomplex of the closed cover Y = Vini=1 is a quasi-isomorphism.

Proposition 3.3.5. Let Y• be a simplicial variety. Then the system of allproper hypercovers of Y• is filtered up to simplicial homotopy. It is functorial inY•. The subsystem of smooth proper hypercovers is cofinal.


Proof. The first statement is [SGA4.2], Expose V, Theoreme 7.3.2. For thesecond assertion, it suffices to construct a smooth proper hypercover for anyY•. Recall that by Hironaka’s resolution of singularities [Hi1], or by de Jong’stheorem on alterations [dJ], we have for any variety Y a proper surjective mapX → Y with X smooth. By the technique of [SGA4.2], Expose Vbis, Proposi-tion 5.1.3 (see also [D5] 6.2.5), this allows to construct X•.

3.3.3 Definition of de Rham cohomology in the generalcase

Let again k be a field of characteristic 0.

Definition 3.3.6. Let X be a variety over k and X• → X a smooth properhypercover. Let C(X•) ∈ ZSm be the associated complex We define algebraicde Rham cohomology of X by

HidR(X) = Hi (RΓdR(X•))

with RΓdR as in Definition 3.1.7. Let D ⊂ X be a closed subvariety and D• → Da smooth proper hypercover such that there is a commutative diagram

D• −−−−→ X•y yD −−−−→ X

We define relative algebraic de Rham cohomology of the pair (X,D) by

HidR(X,D) = Hi (Cone(RΓ(X•)→ RΓ(D•))[−1]) .

Proposition 3.3.7. Algebraic de Rham cohomology is a well-defined functor,independent of the choice of hypercoverings of X and D.

Remark 3.3.8. RΓdR defines a functor

Var→ K+(k−Vect)

but not to C+(k−Vect). Hence it does not extend directly to Cb(Q[Var]). Weavoid addressing this point by the use of the h-topology instead.

Proof. This is a special case of descent for h-covers and hence a consequence ofProposition 3.2.4.

Alternatively, we can deduce if from the case of singular cohomology. Recallthat algebraic de Rham cohomology is well-behaved with respect to extensionsof the ground field. Without loss of generality, we may assume that k is finitelygenerated over Q and hence embeds into C. Then we apply the period iso-morphism of Definition 5.3.1. It remains to check the analogue for singularcohomology. This is Theorem 2.7.6.


Example 3.3.9. Let X be a smooth affine variety and D a simple divisor withnormal crossings. Let D1, . . . , Dn be the irreducible components. Let X• be theconstant simplicial variety X and D• as in Example 3.3.4. Then algebraic deRham cohomology D is computed by the total complex of the double complex(Di0,...,ip being the (p+ 1)-fold intersection of components)

Kp,q =⊕

i0<···<ip

ΩqDi0,...,ip

(Di0,...,ip

)with differential dp,q =

∑pj=0(−1)j∂∗j the Cech differential and δp,q differentia-

tion of differential forms.

Relative algebraic de Rham cohomology of (X,D) is computed by the totalcomplex of the double complex

Lp,q =

Kp−1,q p > 0,

ΩqX(X) p = 0.

Remark 3.3.10. Establishing the expected properties of relative algebraic deRham cohomology is lengthy. Particularly complicated is the handling of themultiplicative structure which uses the the functor between complexes in Z[Sm]and simplicial objects in Z[Sm] and the product for simplicial objects. We donot go into the details but rely on the comparison with h-cohomology instead.

3.3.4 Hartshorne’s method

We want to review Hartshorne’s definition from [Ha1]. As before let k be a fieldof characteristic 0.

Definition 3.3.11. Let X be a smooth variety over k, i : Y ⊂ X a closedsubvariety. We define algebraic de Rham cohomology of Y as

HiH−dR(Y ) = Hi(X, Ω•X),

where X is the formal completion of X along Y and Ω•X the formal completionof the complex of algebraic differential forms on X.

Proposition 3.3.12 ([Ha1] Theorem (1.4)). Let Y be as in Definition 3.3.11.Then Hi

H−dR(Y ) is independent of the choice of X. In particular, if Y is smooth,the definition agrees with the one in Definition 3.1.2.

Theorem 3.3.13. The three definition of algebraic de Rham cohomology (Def-inition 3.3.6 via hypercovers, Definition 3.3.11 via embedding into smooth vari-eties, Definition 3.2.3 using the h-topology) agree.

Proof. The comparison of HiH−dR(X) and Hi

dR(Xeh) is [Ge], Theorem 4.10. Itagrees with Hi

dR(Xh) by [HJ], Proposition 6.1. By [HJ], Proposition 7.4 it agreesalso with the definition via hypercovers.


3.3.5 Using geometric motives

In Chapter 10 we are going to introduce the triangulated category of effectivegeometric motives DM eff

gm over k with coefficients in Q. We only review themost important properties here and refer to Chapter 10 for more details. Fortechnical reasons, it is easier to work with the affine version.

The objects in DM effgm are the same as the objects in Cb(SmCor) where SmCor

is the category of correspondences, see Section 1.1 and we denote SmCorAff thefull subcategory with objects smooth affine varieties.

Lecomte and Wach in [LW] explain how to define an operation of correspon-dences on Ω•X(X). We give a quick survey of their method.

For any normal variety Z let Ωp,∗∗Z be the OZ-double dual of the sheaf of p-differentials. This is nothing but the sheaf of reflexive differentials on Z.

If Z ′ → Z is a finite morphism between normal varietes which is genericallyGalois with covering group G, then by [Kn]

Ωp,∗∗Z (Z) ∼= Ωp,∗∗Z′ (Z ′)G .

Let X and Y be smooth affine varieties. Assume for simplicity that X and Yare connected. Let Γ ∈ Cor(X,Y ) be a prime correspondence, i.e., Γ ⊂ X × Yan integral closed subvariety which is finite and dominant over X. Choose afinite Γ → Γ such that Γ is normal and the covering Γ → X generically Galoiswith covering group G. In this case, X = Γ/G.

Definition 3.3.14. For a correspondence Γ ∈ Cor(X,Y ) as above, we define

Γ∗ : Ω•Y (Y )→ Ω•X(X)

as the composition

Ω•Y (Y )→ Ω•Γ(Γ)→ Ω•,∗∗

Γ(Γ)

1|G|

∑g∈G g

∗

−−−−−−−−→ Ω•,∗∗Γ

(Γ)G = Ω•X(X) .

This is well-defined and compatible with composition of correspondences. Wecan now define de Rham cohomology for complexes of correspondences.

Definition 3.3.15. Let X• ∈ Cb(SmCorAff). We define

RΓdR(X•) = TotRΓdR(Xn)n∈Z .

andHi

dR(X•) = HiRΓdR(X•) .

Note that there is a simple functor SmAff → SmCor. It assigns an object toitself and a morphism to its graph. This induces

i : Cb(Q[SmAff])→ DM effgm .


By construction,f∗ = Γ∗f : Ω•Y (Y )→ Ω•X(X)

for any morphism f : X → Y between smooth affine varieties. Hence,

RΓdR(X•) = RΓdR(i(X•)),

where the left hand side was defined in Definition 3.1.7.

Proposition 3.3.16 (Voevodsky). The functor i extends naturally to a functor

i : Cb(Q[Var])→ DM effgm .

Proof. The category of geometric motives constructed from affine varieties onlyagrees with the original DM eff

gm. For details, see [Ha].

The extension to all varieties is a highly non-trivial result of Voevodsky. By[VSF], Chapter V, Corollary 4.1.4, there is functor

Var→ DMgm .

Indeed, the functorX 7→ C∗L(X)

of loc. cit., Section 4.1, which assigns to every variety a homotopy invariantcomplex of Nisnevich sheaves, extends to Cb(Z[Var]) by taking total complexes.We consider it in the derived category of Nisnevich sheaves. Then the functorfactors via the homotopy category Kb(Z[Var]).

By induction on the length of the complex, it follows from the result quotedabove that C∗L(·) takes values in the full subcategory of geometric motives.

Definition 3.3.17. Let D ⊂ X be a closed immersion of varieties. We define

HidR(X,D) = HiRΓdR(i([D → X]) ,

where [D → X] ∈ Cb(Z[Var]) is concentrated in degrees −1 and 0.

Proposition 3.3.18. This definition agrees with the one given in Definition3.3.6.

Proof. The easiest way to formulate the proof is to invoke another variant ofthe category of geometric motives. It does not need transfers, but imposes h-descent instead. Scholbach [Sch1, Definition 3.10] defines the category DM eff

gm,h

as the localization of K−(Q[Var]) with respect to the triangulated subcategorygenerated by complexes of the form X × A1 → X and h-hypercovers X• → Xand closed under certain infinite sums. By definition of DM eff

gm,h, any hypercov-

ering X• → X induces an isomorphism of the associated complexes in DM effgm,h.

By resolution of singularities, any object of DM effgm,h is isomorphic to an ob-

ject where all components are smooth. Hence we can replace K−(Q[Var]) by


K−(Q[Sm]) in the definition without any change. We have seen how algebraic deRham cohomology is defined on K−(Q[Sm]). By homotopy invariance (Corol-lary 3.1.10) and h-descent of the de Rham complex (Proposition 3.3.7), thedefinition of algebraic de Rham cohomology factors via DM eff

gm,h.

This gives a definition of algebraic de Rham cohomology for K−(Q[Var]) whichby construction agrees with the one in Definition 3.3.6. On the other hand,the main result of [Sch1] is that DM eff

gm can be viewed as full subcategory of

DM effgm,h. This inclusion maps the motive of a (possibly singular) variety to the

motive of a variety. As the two definitions of algebraic de Rham cohomology ofmotives agree on motives of smooth varieties, they agree on all motives.

3.3.6 The case of divisors with normal crossings

We are going to need the following technical result in order to give a simplifieddescription of periods.

Proposition 3.3.19. Let X be a smooth affine variety of dimension d andD ⊂ X a simple divisor with normal crossings. Then every class in Hd

dR(X,D)is represented by some ω ∈ ΩdX(X).

The proof will be given at the end of this section.

Let D = D1 ∪ · · · ∪Dn be the decomposition into irreducible components. ForI ⊂ 1, . . . , n, let again

DI =⋂i∈I

Di .

Recall from Example 3.3.9 that the de Rham cohomology of (X,D) is computedby the total complex of

Ω•X(X)→n⊕i=1

Ω•Di(Di)→⊕i<j

Ω•Di,j (Di,j)→ · · · → Ω•D1,2,...,n(D1,2,...,n) .

Note that DI has dimension d− |I|, hence the double complex is concentratedin degrees p, q ≥ 0, p+ q ≤ d. By definition, the classes in the top cohomologygroup Hd

dR(X,D) are presented by a tuple

(ω0, ω1, . . . , ωn) ω0 ∈ ΩdX(X), ωi ∈⊕|I|=i

Ωd−iDI(DI) , i > 0 .

All such tuples are cocycles for dimension reasons. We have to show that,modulo coboundaries, we can assume ωi = 0 for all i > 0.

Lemma 3.3.20. The maps

Ωd−1X (X)→

n⊕i=1

Ωd−1Di

(Di)⊕|I|=s

Ωd−s−1DI

(DI)→⊕|J|=s+1

Ωd−s−1DJ

(DJ)


are surjective.

Proof. X and all Di are assumed affine, hence the global section functor isexact. It suffices to check the assertion for the corresponding sheaves on X andhence locally for the etale topology. By replacing X by an etale neighbourhoodof a point, we can assume that there is a global system of regular paramterst1, . . . , td on X such that Di = ti = 0 for i = 1, . . . , n. First consider the cases = 0. The elements of Ωd−1

Di(Di) are locally of the form fidt1∧· · ·∧ ˙dti∧· · ·∧ td

(omitting the factor at i). Again by replacing X by an open subvariety, we canassume they are globally of this shape. The forms can all be lifted to X.

ω =

n∑i=1

fidt1 ∧ · · · ∧ ˙dti ∧ · · · ∧ td

is the preimage we were looking for.

For s ≥ 1 we argue by induction on d and n. If n = 1, there is nothing to show.This settles the case d = 1. If n > 0, consider the decomposition

0 0y y⊕|I|=s,I⊂1,...,n−1

Ωd−s−1DI

(DI) −−−−→⊕

|J|=s+1,J⊂1,...,n−1Ωd−s−1DJ

(DJ)y y⊕|I|=s,I⊂1,...,n

Ωd−s−1DI

(DI) −−−−→⊕

|J|=s+1,J⊂1,...,nΩd−s−1DJ

(DJ)y y⊕|I|=s,I⊂1,...,n,n∈I

Ωd−s−1DI

(DI) −−−−→⊕

|J|=s+1,J⊂1,...,n,n∈JΩd−s−1DJ

(DJ)y y0 0

The arrow on the top is surjective by induction on n. The arrow on the bottomreproduces the assertion for X replaced by Dn and D replaced by Dn ∩ (D1 ∪· · · ∪ Dn−1). By induction, it is surjective. Hence, the arrow in the middle issurjective.

Proof of Proposition 3.3.19. Consider a cocycle ω = (ω0, ω1, . . . , ωn) as explainedabove. We argue by descending induction on the degree i. Consider ωn ∈⊕|I|=n Ωd−nDI

(DI). By the lemma, there is

ω′n−1 ∈⊕|I|=n−1

Ωd−nDI(DI)


such that ∂ω′n−1 = ωn. We replace ω by ω ± dω′n−1 (depending on the signs inthe double complex). By construction, its component in degree n vanishes.

Hence, without loss of generality, we have ωn = 0. Next, consider ωn−1 etc.

Chapter 4

Holomorphic de Rhamcohomology

We are going to define a natural comparison isomorphism between de Rhamcohomology and singular cohomology of varieties over the complex numbers.The link is provided by holomorphic de Rham cohomology which we study inthis chapter.

4.1 Holomorphic de Rham cohomology

Everything we did in the algebraic setting also works for complex manifolds,indeed this is the older notion.

We write OholX for the sheaf of holomorphic functions on a complex manifold X.

4.1.1 Definition

Definition 4.1.1. Let X be a complex manifold. Let Ω1X be the sheaf of

holomorphic differentials on X. For p ≥ 0 let

ΩpX = ΛpΩ1X

be the exterior power in the category of OholX -modules and (Ω•X , d) the holomor-

phic de Rham complex.

The differential is defined as in the algebraic case, see Definition 3.1.1.

Definition 4.1.2. Let X be a complex manifold. We define holomorphic deRham cohomology of X as hypercohomology

HidRan(X) = Hi(X,Ω•X) .

109

110 CHAPTER 4. HOLOMORPHIC DE RHAM COHOMOLOGY

As in the algebraic case, de Rham cohomology is a contravariant functor. Theexterior products induces a cup-product.

Proposition 4.1.3 (Poincare lemma). Let X be a complex manifold. Thenatural map of sheaves C→ Ohol

X induces an isomorphism

Hising(X,C)→ Hi

dRan(X) .

Proof. By Theorem 2.2.5, we can compute singular cohomology as sheaf coho-mology on X. It remains to show that the complex

0→ C→ OholX → Ω1

X → Ω2X → . . .

is exact. Let ∆ be the unit ball in C. The question is local, hence we mayassume that X = ∆d. There is a natural isomorphism

Ω•∆d∼= (Ω•∆)

⊗d

Hence it suffices to treat the case X = ∆. In this case we consider

0→ C→ Ohol(∆)→ Ohol(∆)dt→ 0 .

The elements of Ohol(∆) are of the form∑i≥0 ait

i with radius of convergence1. The differential has the form∑

i≥0

aiti 7→

∑i≥0

iaiti−1dt .

The kernel is given by the constants. It is surjective because the antiderivativehas the same radius of convergence as the original power series.

Proposition 4.1.4 (Kunneth formula). Let X,Y be complex manifolds. Thereis a natural multiplication induced from wedge product of differential forms

HidR(X)⊗k Hj

dR(Y )→ Hi+jdR (X × Y ) .

It induces an isomorphism

HndR(X × Y ) ∼=

⊕i+j=n

HidR(X)⊗k Hj

dR(Y ) .

Proof. The construction of the morphism is the same as in the algebraic case,see Proposition 3.1.9. The quasi-isomorphism C → Ω• is compatible with theexterior products. Hence the isomorphism reduces to the Kunneth isomorphismfor singular cohomology, see Proposition 2.4.1.

4.1. HOLOMORPHIC DE RHAM COHOMOLOGY 111

4.1.2 Holomorphic differentials with log poles

Let j : X → X be a an open immersion of complex manifolds. Assume thatD = X r X is a divisor with normal crossings, i.e., locally on X there isa coordinate system (t1, . . . , tn) such that D is given as the set of zeroes oft1t2 . . . tr with 0 ≤ r ≤ n.

Definition 4.1.5. LetΩ1X〈D〉 ⊂ j∗Ω

1X

be the locally free OX -module with the following basis: if U ⊂ X is an openwith coordinates t1, . . . , tn and D|U given by the equation t1 . . . tr = 0, thenΩ1X〈D〉|U has Ohol

X-basis

dt1t1, . . . ,

dtrtr, dtr+1, . . . , dtn .

For p > 1 letΩpX〈D〉 = ΛpΩ1

X〈D〉 .

We call the Ω•X〈D〉 the complex of differentials with log poles along D.

Note that the differential of j∗Ω•X respects Ω•

X〈D〉, so that this is indeed a

complex.

Proposition 4.1.6. The inclusion induces a natural isomorphism

Hi(X,Ω•X〈D〉)→ Hi(X,Ω•X) .

This is [D4] Proposition 3.1.8. The algebraic analogue was treated in Proposi-tion 3.1.16.

Proof. Note that j : X → X is Stein, hence j∗ is exact and we have

Hi(X,Ω•X) ∼= Hi(X, j∗Ω•X) .

It remains to show thatι : Ω•X〈D〉 → j∗Ω

•X

is a quasi-isomorphism, or, equivalently, that Coker(ι) is exact. The statementis local, hence we may assume that X is a coordinate ball and D = V (t1 . . . tr).We consider the stalk in 0. The complexes are tensor products of the complexesin the 1-dimensional situation. Hence it suffices to consider the case n = 1. Ifr = 0, then there is nothing to show.

In remains to consider the following situation: let Ohol be ring of germs ofholomorphic functions at 0 ∈ C and Khol the ring of germs of holomorphicfunctions with an isolated singularity at 0. The ring Ohol is given by powerseries with a positive radius of convergence. The field Khol is given by Laurent


series converging on some punctured neighborhood t | 0 < t < ε. We have tocheck that the complex

Khol/Ohol → (Khol/t−1Ohol)dt

is acyclic.

We pass to the principal parts. The differential has the form∑i>0

ait−i 7→

∑i>0

(−i)ait−i−1

It is obviously injective. For surjectivity, note that the antiderivative∫:∑i>1

bit−i 7→

∑i>1

bi−i+ 1

t−i+1

maps convergent Laurent series to convergent Laurent series.

4.1.3 GAGA

We work over the field of complex numbers.

An affine variety X ⊂ AnC is also a closed set in the analytic topology on Cn. IfX is smooth, the associated analytic space Xan in the sense of Section 1.2.1 isa complex submanifold. As in loc. cit., we denote by

α : (Xan,OholXan)→ (X,OX)

the map of locally ringed spaces. Note that any algebraic differential form isholomorphic, hence there is a natural morphism of complexes

α−1Ω•X → Ω•Xan .

It induces

α∗ : HidR(X)→ Hi

dRan(Xan) .

Proposition 4.1.7 (GAGA for de Rham cohomology). Let X be a smoothvariety over C. Then the natural map

α∗ : HidR(X)→ Hi

dRan(Xan)

is an isomorphism.

If X is smooth and projective, this is a standard consequence of Serre’s com-parison result for cohomology of coherent sheaves (GAGA). We need to extendthis to the open case.

4.2. DE RHAM COHOMOLOGY VIA THE H′-TOPOLOGY 113

Proof. Let j : X → X be a compactification such that D = X rX is a simpledivisor with normal crossings. The change of topology map α also induces

α−1j∗Ω•X → jan

∗ Ω•Xan

which respects differential with log-poles

α−1ΩX•〈D〉 → jan∗ Ω•Xan〈Dan〉 .

Hence we get a commutative diagram

HidR(X) −−−−→ Hi

dRan(Xan)x xHi(X,Ω•

X〈D〉) −−−−→ Hi(Xan,Ω•

Xan〈Dan〉)

By Proposition 3.1.16 in the algebraic, and Proposition 4.1.6 in the holomorphiccase, the vertical maps are isomorphism. By considering the Hodge to de Rhamspectral sequence (attached to the stupid filtration on Ω•X〈D〉), it suffices toshow that

Hp(X,ΩqX〈D〉)→ Hp(Xan,Ωq

Xan〈Dan〉)

is an isomorphism for all p, q. Note that X is smooth, projective and ΩqX〈D〉 is

coherent. Its analytification α−1ΩqX〈D〉⊗α−1OXO

holXan is nothing but Ωq

Xan〈Dan〉.By GAGA [Se1], we have an isomorphism in cohomology.

4.2 De Rham cohomology via the h′-topology

We address the singular case via the h′-topology on (An/X) introduced in Def-inition 2.7.1.

4.2.1 h′-differentials

Definition 4.2.1. Let Ωph′ be the h′-sheafification of the presheaf

Y 7→ ΩpY (Y )

on the category of complex analytic spaces An.

Theorem 4.2.2 (Jorder [Joe]). Let X be a complex manifold. Then

ΩpX(X) = Ωph′(X) .

Proof. Jorder defines in [Joe, Definition 1.4.1] what he calls h-differentials Ωphas the presheaf pull-back of Ωp from the category of manifolds to the categoryof complex analytic spaces. (There is no mention of a topology in loc.cit.) In


[Joe, Proposition 1.4.2 (4)] he establishes that Ωph(X) = ΩpX(X) in the smoothcase. It remains to show that Ωph = Ωph′ . By resolution of singularities, everyX is smooth locally for the h′-topology. Hence it suffices to show that Ωph isan h′-sheaf. By [Joe, Lemma 1.4.5], the sheaf condition is satisfied for propercovers. The sheaf condition for open covers is satisfied because already ΩpX is asheaf in the ordinary topology.

Lemma 4.2.3 (Poincare lemma). Let X be a complex analytic space. Then thecomplex

Ch′ → Ω•h′

of h′-sheaves on (An/X)h′ is exact.

Proof. We may check this locally in the h′-topology. By resolution of singuaritiesit suffices to consider sections over some Y which is smooth and even an openball in Cn. By Theorem 4.2.2 the complex reads

C→ Ω•Y (Y ) .

By the ordinary holomorphic Poincare Lemma 4.1.3, it is exact.

Remark 4.2.4. The main topic of Jorder’s thesis [Joe] is to treat the questionof a Poincare Lemma for h′-forms with respect to the usual topology. This ismore subtle and fails in general.

4.2.2 De Rham cohomology

We now turn to de Rham cohomology.

Definition 4.2.5. Let X be a complex analytic space.

1. We define h′-de Rham cohomology as hypercohomology

HidRan(Xh′) = Hi

h′((Sch/X)h′ ,Ω•h′) .

2. Let i : Z → X a closed subspace. Put

Ωph/(X,Z) = Ker(Ωph/X → i∗Ωph/Z)

in the category of abelian sheaves on (An/X)h′ .

We define relative h′-de Rham cohomology as

HpdRan(Xh′ , Zh′) = Hp

h′((An/X)h′ ,Ω∗h/(X,Z)) .

Lemma 4.2.6. The properties (long exact sequence, excision, Kunneth for-mula) of relative algebraic H-de Rham cohomology (see Section 3.2) are alsosatisfied in relative h′-de Rham cohomology.

4.2. DE RHAM COHOMOLOGY VIA THE H′-TOPOLOGY 115

Proof. The proofs are the same as Section 3.2, respectively in [HJ, Section 7.3].The proof relies on the computation of Ωph′(D) when D is a normal crossingsspace. Indeed, the same argument as in the proof of [HJ, Proposition 4.9] showsthat

Ωph′(D) = ΩpD(D)/torsion .

As in the previous case, exterior multiplication of differential forms induces aproduct structure on h′-de Rham cohomology.

Corollary 4.2.7. For all X ∈ An and closed immersions i : Z → X theinclusion of the Poincare lemma induces a natural isomorphism

Hising(X,Z,C)→ Hi

dRan(Xh′ , Zh′) ,

compatible with long exact sequences and multiplication. Moreover, the naturalmap

HidRan(Xh′)→ Hi

dRan(X)

is an isomorphism if X is smooth.

Proof. By the Poincare Lemma 4.2.3, we have a natural isomorphism

Hih′(Xh′ , Zh′ ,Ch′)→ Hi

dRan(Xh′ , Zh′) .

We combine it with the comparison isomorphism with singular cohomology ofProposition 2.7.4.

The second statement holds because both compute singular cohomology byProp. 2.7.4 and Prop. 4.1.3.

4.2.3 GAGA

We work over the base field C. As before we consider the analytification functor

X 7→ Xan

which takes a separated scheme of finite type over C to a complex analytic space.We recall the map of locally ringed spaces

α : Xan → X .

We want to view it as a morphism of topoi

α : (An/Xan)h′ → (Sch/X)h .

Definition 4.2.8. Let X ∈ Sch/C. We define the h′-topology on the category(Sch/X)h′ to be the smallest Grothendieck topology such that the following arecovering maps:



2. open covers.

If F is a presheaf of An/X, we denote by Fh′ its sheafification in the h′-topology.

Lemma 4.2.9. 1. The morphism of sites (Sch/X)h → (Sch/X)h′ inducesan isomorphism on the categories of sheaves.

2. The analytification functor induces a morphism of sites

(An/Xan)h′ → (Sch/X)h′ .

Proof. By [Voe] Theorem 3.1.9 any h-cover can be refined by a cover in normalform which is a composition of open immersions followed by proper maps. Thisshows the first assertion. The second is clear by construction.

By h′-sheafifiying, the natural morphism of complexes

α−1Ω•X → Ω•Xan

of Section 4.1.3, we also obtain

α−1Ω•h → Ω•h′

on (An/Xan)h′ . It induces

α∗ : HidR(Xh)→ Hi

dRan(Xanh′ ) .

Proposition 4.2.10 (GAGA for h′-de Rham cohomology). Let X be a varietyover C and Z a closed subvariety. Then the natural map

α∗ : HidR(Xh, Zh)→ Hi

dRan(Xanh′ , Z

anh′ )

is an isomorphism. It is compatible with long exact sequences and products.

Proof. By naturality, the comparison morphism is compatible with long exactsequences. Hence it suffices to consider the absolute case.

Let X• → X be a smooth proper hypercover. This is a cover in h′-topology,hence we may replace X by X• on both sides. As all components of X•are smooth, we may replace h-cohomology by Zariski-cohomology in the al-gebraic setting (see Proposition 3.2.4). On the analytic side, we may replaceh′-cohomology by ordinary sheaf cohomology (see Corollary 2.7.4). The state-ment then follows from the comparison in the smooth case, see Proposition4.1.7.

Chapter 5

The period isomorphism

The aim of this section is to define well-behaved isomorphisms between singularand de Rham cohomology of algebraic varieties.

5.1 The category (k,Q)−Vect

We introduce a simple linear algebra category which will later allow to formalizethe notion of periods. Throughout, let k ⊂ C be a subfield.

Definition 5.1.1. Let (k,Q)−Vect be the category of triples (Vk, VQ, φC) whereVk is a finite dimensional k-vector space, VQ a finite dimensional Q-vector spaceand

φC : Vk ⊗k C→ VQ ⊗Q Ca C-linear isomorphism. Morphisms in (k,Q)−Vect are linear maps on Vk andVQ compatible with comparison isomorphisms.

Note that (k,Q)−Vect is a Q-linear additive tensor category with the obviousnotion of tensor product. It is rigid, i.e., all objects have strong duals. It is evenTannakian with projection to the Q-component as fibre functor.

For later use, we make the duality explicit:

Remark 5.1.2. Let V = (Vk, VQ, φC) ∈ (k,Q)−Vect. The the dual V ∨ is givenby

V ∨ = (V ∗k , V∗Q , (φ

∗)−1

where ·∗ denotes the vector space dual over k and Q or C. Note that the inverseis needed in order to make the map go in the right direction.

Remark 5.1.3. The above is a simplification of the category of mixed Hodgestructures introduced by Deligne, see [D4]. It does not take the weight andHodge filtration into account. In other words: there is a faithful forgetful functorfrom mixed Hodge structures over k to (k,Q)−Vect.

117

118 CHAPTER 5. THE PERIOD ISOMORPHISM

Example 5.1.4. The invertible objects are those where dimk Vk = dimQ VQ =1. Up to isomorphism they are of the form

L(α) = (k,Q, α) with α ∈ C∗ .

5.2 A triangulated category

We introduce a triangulated category with a t-structure whose heart is (k,Q)−Vect.

Definition 5.2.1. A cohomological (k,Q)−Vect-complex consists of the follow-ing data:

• a bounded below complex K•k of k-vector spaces with finite dimensionalcohomology;

• a bounded below complex K•Q of Q-vector spaces with finite dimensionalcohomology;

• a bounded below complex K•C of C-vector spaces with finite dimensionalcohomology;

• a quasi-isomorphism φk,C : K•k ⊗k C→ K•C;

• a quasi-isomorphism φQ,C : K•Q ⊗Q C→ K•C.

Morphisms of cohomological (k,Q)−Vect-complexes are given by a pair of mor-phisms of complexes on the k-, Q- and C-component such that the obviousdiagram commutes. We denote the category of cohomological (k,Q)−Vect-complexes by C+

(k,Q).

Let K and L be objects of C+(k,Q). A homotopy between K and L is a homotopy

in the k-, Q- and C-component compatible under the comparison maps. Twomorphisms in C+

(k,Q) are homotopic if they differ by a homotopy. We denote by

K+(k,Q) the homotopy category of cohomological (k,Q)−Vect-complexes.

A morphism inK+(k,Q) is called quasi-isomorphism if its k-, Q-, and C-components

are quasi-isomorphisms. We denote by D+(k,Q) the localization of K+

(k,Q) with re-

spect to quasi-isomorphisms. It is called the derived category of cohomological(k,Q)−Vect-complexes.

Remark 5.2.2. This is a simplification of the category of mixed Hodge com-plexes introduced by Beilinson [Be2]. A systematic study of this type of categorycan be found in [Hu1, §4]. In the language of loc.cit., it is the rigid glued cate-gory of the category of k-vector spaces and the category of Q-vector spaces viathe category of C-vector spaces and the extension of scalars functors. Note thatthey are exact, hence the construction simplifies.

5.3. THE PERIOD ISOMORPHISM IN THE SMOOTH CASE 119

Lemma 5.2.3. D+(k,Q) is a triangulated category. It has a natural t-structure

with

Hi : D+(k,Q) → (k,Q)−Vect

defined componentwise. The heart of the t-structure is (k,Q)−Vect.

Proof. This is straightforward. For more details see [Hu1, §4].

Remark 5.2.4. In [Hu1, 4.2, 4.3], explicit formulas are given for the mor-phisms in D+

(k,Q). The category has cohomological dimension 1. For K,L ∈(k,Q)−Vect, the group HomD+

(k,Q)(K,L[1]) is equal to the group of Yoneda ex-

tensions. As in [Be2], this implies that D+(k,Q) is equivalent to the bounded

derived category D+((k,Q)−Vect). We do not spell out the details because weare not going to need these properties.

There is an obvious definition of a tensor product on C+(k,Q). LetK•, L• ∈ C+

(k,Q).

We define K• ⊗ L• with k,Q,C-component given by the tensor product ofcomplexes of vector spaces over k, Q, and C, respectively (see Example 1.3.4).Tensor product of two quasi-isomorphisms defines the comparison isomorphismon the tensor product.

It is associative and commutative. Note that the

Lemma 5.2.5. C+(k,Q), K

+(k,Q) and D+

(k,Q) are associative and commutative ten-

sor categories with the above tensor product. The cohomology functor H∗ com-mutes with ⊗. For K•, L• in D+

(k,Q), we have a natural isomorphism

H∗(K•)⊗H∗(L•)→ H∗(K• ⊗ L•).

It is compatible with the associativity constraint. It is compatible with the com-mutativity constraint up to the sign (−1)pq on Hp(K•)⊗Hq(L•).

Proof. The case of D+(k,Q) follows immediately from the case of complexes of

vector spaces, where it is well-known. The signs come from the signs in thetotal complex of a bicomplex, in this case, tensor product of complexes, seeSection 1.3.3.

Remark 5.2.6. This is again simpler than the case treated in [Hu1, Chap-ter 13], because we do not need to control filtrations and because our tensorproducts are exact.

5.3 The period isomorphism in the smooth case

Let k be a subfield of C. We consider smooth varieties over k and the complexmanifold Xan associated to X ×k C.


Definition 5.3.1. Let X be a smooth variety over k. We define the periodisomorphism

per : H•dR(X)⊗k C→ H•sing(X,Q)⊗Q C

to be the isomorphism given by the composition of the isomorphisms

1. H•dR(X)⊗k C→ H•dR(X ×k C) of Lemma 3.1.11,

2. H•dR(X ×k C)→ H•dRan(Xan) of Proposition 4.1.7,

3. the inverse of H•dRan(Xan)→ H•sing(Xan,C) of Proposition 4.1.3,

4. the inverse of the change of coefficients isomorphism H•sing(Xan,C) →H•sing(Xan,Q)⊗Q C.

We define the period pairing

per : H•dR(X)×Hsing• (Xan,Q)→ C

to be the map

(ω, γ) 7→ γ(per(ω))

where we view classes in singular homology as linear forms on singular coho-mology.

Recall the category (k,Q)−Vect introduced in Section 5.1.

Lemma 5.3.2. The assignment

X 7→ (H•dR(X), H•sing(X),per)

defines a functor

H : Sm→ (k,Q)−Vect .

For all X,Y ∈ Sm, the Kunneth isomorphism induces an natural isomorphism

H(X)⊗H(Y )→ H(X × Y ) .

The image of H is closed under direct sums and tensor product.

Proof. Functoriality holds by construction. The Kunneth morphism is inducedfrom the Kunneth isomorphism in singular cohomology (Proposition 2.4.1) andalgebraic de Rham cohomology (see Proposition 3.1.9). All identifications inDefinition 5.3.1 are compatible with the product structure. Hence we havedefined a Kunneth morphism in H. It is an isomorphism because it is an iso-morphism in singular cohomology.

The direct sum realized by the disjoint union. The tensor product is realizedby the product.

5.4. THE GENERAL CASE (VIA THE H′-TOPOLOGY) 121

In Chapter 9, we are going to study systematically the periods of the objects inH(Sm).

The period isomorphism has an explicit description in terms of integration.

Theorem 5.3.3. Let X be a smooth affine variety over k and ω ∈ Ωi(X) a

closed differential form with de Rham class [ω]. Let c ∈ Hsingd (Xan,Q) be a sin-

gular homology class. Let∑ajγj with aj ∈ Q and γj : ∆i → Xan differentiable

singular cycles as in Definition 2.2.2. Then

per([ω], c) =∑

aj

∫∆i

γ∗(ω) .

Remark 5.3.4. We could use the above formula as a definition of the periodpairing, at least in the affine case. By Stokes’ theorem, the value only dependson the class of ω.

Proof. Let Ai(Xan) be group of C-valued C∞-differential forms and AiXan theassociated sheaf. By the Poincare lemma and its C∞-analogue the morphisms

C→ Ω•Xan → A•Xan

are quasi-isomorphism. It induces a quasi-isomorphism

Ω•Xan(Xan)→ A•(Xan)

because both compute singular cohomology in the affine case. Hence it suf-fices to view ω as a C∞-differential form. By the Theorem of de Rham, see[Wa], Sections 5.34-5.36, the period isomorphism is realized by integration oversimplices.

Example 5.3.5. For X = Pnk , we have

H2j(Pnk ) = L((2πi)j)

with L(α) the invertible object of Example 5.1.4.

5.4 The general case (via the h′-topology)

We generalize the period isomorphism to relative cohomology of arbitrary vari-eties.

Let k be a subfield of C. We consider varieties over k and the complex analyticspace Xan associated to X ×k C.

Definition 5.4.1. Let X be a variety over k, and Z ⊂ X a closed subvariety.We define the period isomorphism

per : H•dR(X,Z)⊗k C→ H•sing(X,Z,Q)⊗Q C

to be the isomorphism given by the composition of the isomorphisms


1. H•dR(X,Z)⊗k C→ H•dR(X ×k C, Z ×k C) of Lemma 3.2.14,

2. H•dR(X ×k C, Z ×k C)→ H•dRan(Xanh′ , Z

anh′ ) of Proposition 4.2.10,

3. the inverse of H•dRan(Xanh′ , Z

anh′ )→ H•sing(Xan, Zan,C) of Corollary 4.2.7,

4. the inverse of the change of coefficients isomorphism H•sing(Xan, Zan,C)→H•sing(Xan, Zan,Q)⊗Q C.


per : H•dR(X,Z)×Hsing• (Xan, Zan,Q)→ C

to be the map

(ω, γ) 7→ γ(per(ω)),

where we view classes in singular homology as linear forms on singular coho-mology.

Lemma 5.4.2. The assignment

(X,Z) 7→ (H•dR(X,Z), H•sing(X,Z),per)

defines a functor denoted H on the category of pairs X ⊃ Z with values in(k,Q)−Vect. For all Z ⊂ Z, Z ′ ⊂ X ′, the Kunneth isomorphism induces anatural isomorphism

H(X,Z)⊗H(X ′, Z ′)→ H(X ×X ′, X × Z ′ ∪ Z ×X ′) .

The image of H is closed under direct sums and tensor product.

If Z ⊂ Y ⊂ X is a triple, the there is a induced long exact sequence in(k,Q)−Vect.

· · · → Hi(X,Y )→ Hi(X,Z)→ Hi(Y, Z)∂−→ Hi+1(X,Y )→ . . . .

Proof. Functoriality and compatibility with long exact sequences holds by con-struction. The Kunneth morphism is induced from the Kunneth isomorphismin singular cohomology (Proposition 2.4.1) and algebraic de Rham cohomology(see Proposition 3.1.9). All identifications in Definition 5.3.1 are compatiblewith the product structure. Hence we have defined a Kunneth morphism in H.It is an isomorphism because it is an isomorphism in singular cohomology.

The direct sum realized by the disjoint union. The tensor product is realizedby the product.

5.5. THE GENERAL CASE (DELIGNE’S METHOD) 123

5.5 The general case (Deligne’s method)

We generalize the period isomorphism to relative cohomology of arbitrary vari-eties.

Let k be a subfield of C.

Recall from Section 3.1.2 the functor

RΓdR : Z[Sm]→ C+(k−Mod)

which maps a smooth variety to a natural complex computing its de Rhamcohomology. In the same way, we define using the Godement resolution (seeDefinition 1.4.8)

RΓsing(X) = Γ(Xan, Gd(Q)) ∈ C+(Q−Mod)

a complex computing singular cohomology of Xan. Moreover, let

RΓdRan(X) = Γ(Xan, Gd(Ω•Xan) ∈ C+(C−Mod)

be a complex computing holomorphic de Rham cohomology of Xan.

Lemma 5.5.1. Let X be a smooth variety over k.

1. As before let α : Xan → X ×k C be the morphism of locally ringed spacesand β : X ×k C→ X the natural map. The inclusion α−1β−1Ω•X → Ω•Xan

induces a natural quasi-isomorphism of complexes

φdR,dRan : RΓdR(X)⊗k C→ RΓdRan(X) .

2. The inclusion Q → Ω•Xan induces a natural quasi-isomorphism of com-plexes

φsing,dRan : RΓsing(X)⊗Q C→ RΓdRan(X) .

3. We have

per = H•(φsing,dRan)−1H•(φsing,dRan) : H•dR(X)⊗kC)→ H•sing(Xan,Q) .

Proof. The first assertion follows from applying Lemma 1.4.10 to β and α. Asbefore, we identify sheaves on X ×k C with sheaves on the set of closed pointsof X ×k C. This yields a quasi-isomorphism

α−1β−1GdX(Ω•X)→ GdXan(α−1β−1Ω•X) .

We compose with

GdXan(α−1β−1Ω•X)→ GdXan(Ω•Xan) .


Taking global sections yields by definition a natural Q-linear map of complexes

RΓdR(X)→ RΓdRan(X) .

By extension of scalars we get φdR,dRan . It is a quasi-isomorphism because oncohomology it defines the maps from Lemma 3.1.11 and Proposition 4.1.7.

The second assertion follows from ordinary functoriality of the Godement reso-lution. The last holds by construction.

In other words:

Corollary 5.5.2. The assignment

X 7→ (RΓdR(X), RΓsing(X), RΓdRan(X), φdR,dRan , φsing,dRan)

defines a functor

RΓ : Sm→ C+(k,Q)

where C+(k,Q) is the category of cohomological (k,Q)−Vect-complexes introduced

in Definition 5.2.1.

Moreover,

H•(RΓ(X)) = H(X) ,

where the functor H is defined as above.

Proof. Clear from the lemma.

By naturality, these definitions extend to objects in Z[Sm].


RΓ : K−(ZSm)→ D+(k,Q)

be defined componentwise as the total complex complex of the complex in C+(k,Q).

For X• ∈ C−(ZSm) and i ∈ Z we put

Hi(X•) = HiRΓ(X•) .

Definition 5.5.4. Let k be a subfield of C and X a variety over k with a closedsubvariety D. We define the period isomorphism

per : H•dR(X,D)⊗k C→ H•sing(Xan, Dan)⊗Q C

as follows: let D• → X• be smooth proper hypercovers of D → X as in Defini-tion 3.3.6. Let

C• = ConeC(D•)→ C(X•)) ∈ C−(Z[Sm]) .

5.5. THE GENERAL CASE (DELIGNE’S METHOD) 125

Then H•(RΓ(C•)) consists of

(H•dR(X,D), H•sing(X,D),per) .

In detail: per is given by the composition of the isomorphisms

H•sing(Xan, Dan,Q))⊗Q C→ H•(RΓsing(C•)

withH•φsing,dRan(C•)

−1 H•φdR,dRan(C•) .


per : H•dR(X,D)×Hsing• (Xan, Dan)→ C

to be the map(ω, γ) 7→ γ(per(ω))

where we view classes in relative singular homology as linear forms on relativesingular cohomology.

Lemma 5.5.5. per is well-defined, compatible with products and long exactsequences for relative cohomology.

Proof. By definition of relative algebraic de Rham cohomology (see Definition3.3.6), the morphism takes values in H•dR(X,D) ⊗k C. The first map is anisomorphism by proper descent in singular cohomology, see Theorem 2.7.6.

Compatibility with long exact sequences and multiplication comes from thedefinition.

We make this explicit in the case of a divisor with normal crossings. Recall thedescription of relative de Rham cohomology in this case in Proposition 3.3.19.

Theorem 5.5.6. Let X be a smooth affine variety of dimension d and D ⊂X a simple divisor with normal crossings. Let ω ∈ ΩdX(X) with associatedcohomology class [ω] ∈ Hd

dR(X,D). Let∑ajγj with aj ∈ Q and γj : ∆i → Xan

be a differentiable singular cchain as in Definition 2.2.2 with boundary in Dan.Then

per([ω], c) =∑

aj

∫∆i

γ∗(ω) .

Proof. Let D• as in Section 3.3.6. We apply the considerations of the proofof Theorem 5.3.3 to X and the components of D•. Note that ω|DI = 0 fordimension reasons.


Part II

Nori Motives

127

Chapter 6

Nori’s diagram category

We explain Nori’s construction of an abelian category attached to the repre-sentation of a diagram and establish some properties for it. The constructionis completely formal. It mimicks the standard construction of the Tannakiandual of a rigid tensor category with a fibre functor. Only, we do not have atensor product or even a category but only what we should think of as the fibrefunctor.

The results are due to Nori. Notes from some of his talks are available [N, N1].There is a also a sketch in Levine’s survey [L1] §5.3. In the proofs of the mainresults we follow closely the diploma thesis of von Wangenheim in [vW].

6.1 Main results

6.1.1 Diagrams and representations

Let R be a noetherian, commutative ring with unit.

Definition 6.1.1. A diagram D is a directed graph on a set of vertices V (D)and edges E(D). A diagram with identities is a diagram with a choice of adistinguished edge idv : v → v for every v ∈ D. A diagram is called finite ifit has only finitely many vertices. A finite full subdiagram of a diagram D is adiagram containing a finite subset of vertices of D and all edges (in D) betweenthem.

By abuse of notation we often write v ∈ D instead of v ∈ V (D). The set of alldirected edges between p, q ∈ D is denoted by D(p, q).

Remark 6.1.2. One may view a diagram as a category where composition ofmorphisms is not defined. The notion of a diagram with identity edges is notstandard. The notion is useful later when we consider multiplicative structures.

129

130 CHAPTER 6. NORI’S DIAGRAM CATEGORY

Example 6.1.3. Let C be a small category. Then we can associate a diagramD(C) with vertices the set of objects in C and edges given by morphisms. It iseven a diagram with identities. By abuse of notation we usually also write C forthe diagram.

Definition 6.1.4. A representation T of a diagram D in a small category C isa map T of directed graphs from D to D(C). A representation T of a diagramD with identities is a representation such that id is mapped to id.

For p, q ∈ D and every edge m from p to q we denote their images simply byTp, Tq and Tm : Tp→ Tq (mostly without brackets).

Remark 6.1.5. Alternatively, a representation is defined as a functor from thepath category P(D) to C. Recall that the objects of the path category are thevertices of D, and the morphisms are sequences of directed edges e1e2 . . . enfor n ≥ 0 with the edge ei starting in the end point of ei−1 for i = 2, . . . , n.Morphisms are composed by concatenating edges.

We are particularly interested in representations in categories of modules.

Definition 6.1.6. Let R be a noetherian commutative ring with unit. ByR−Mod we denote the category of finitely generated R-modules. By R−Projwe denote the subcategory of finitely generated projective R-modules.

Note that these categories are essentially small by passing to isomorphic objects,so we will not worry about smallness from now on.

Definition 6.1.7. Let S be a commutative unital R-algebra and T : D →R−Mod a representation. We denote TS the representation

DT−→ R−Mod

⊗RS−−−→ S−Mod .

Definition 6.1.8. Let T be a representation of D in R−Mod. We define thering of endomorphisms of T by

End(T ) :=

(ep)p∈D ∈∏p∈D

EndR(Tp)|eq Tm = Tm ep ∀p, q ∈ D ∀m ∈ D(p, q)

.

Remark 6.1.9. In other words, an element of End(T ) consists of tuples (ep)p∈V (D)

of endomorphisms of Tp, such that all diagrams of the following form commute:

Tp Tq

Tp Tq

ep eq

Tm

Tm

Note that the ring of endomorphisms does not change when we replace D bythe path category P(D).

6.1. MAIN RESULTS 131

6.1.2 Explicit construction of the diagram category

The diagram category can be characterized by a universal property, but it alsohas a simple explicit description that we give first.

Definition 6.1.10 (Nori). Let R be a noetherian commutative ring with unit.Let T be a representation of D in R−Mod.

1. Assume D is finite. Then we put

C(D,T ) = End(T )−Mod

the category of finitely generated R-modules equipped with an R-linearoperation of the algebra End(T ).

2. In general letC(D,T ) = 2−colimFC(F, T |F )

where F runs through the system of finite subdiagrams of D.

More explicitly: the objects of C(D,T ) are the objects of C(F, T |F ) forsome finite subdiagram F . For X ∈ C(F, T |F ) and F ⊂ F ′ we write XF ′

for the image of X in C(F ′, T |F ′). For objects X,Y ∈ C(D,T ), we put

MorC(D,T )(X,Y ) = lim−→F

MorC(F,T |F )(XF , YF ) .

The category C(D,T ) is called the diagram category. With

fT : C(D,T ) −→ R−Mod

we denote the forgetful functor.

Remark 6.1.11. The representation T : D −→ C(D,T ) extends to a functoron the path category P(D). By construction the diagram categories C(D,T )and C(P(D), T ) agree. The point of view of the path category will be usefulChapter 7, in particular in Definition 7.2.1.

In section 6.5 we will prove that under additional conditions for R, satisfied inthe cases of most interest, there is the following even more direct description ofC(D,T ) as comodules over a coalgebra.

Theorem 6.1.12. If the representation T takes values in free modules over afield or Dedekind domain R, the diagram category is equivalent to the categoryof finitely generated comodules (see Definition 6.5.4) over the coalgebra A(D,T )where

A(D,T ) = colimFA(F, T ) = colimFEnd(T |F )∨

with F running through the system of all finite subdiagrams of D and ∨ theR-dual.

The proof of this theorem is given in Section 6.5.


6.1.3 Universal property: Statement

Theorem 6.1.13 (Nori). Let D be a diagram and

T : D −→ R−Mod

a representation of D.

Then there exists an R-linear abelian category C(D,T ), together with a repre-sentation

T : D −→ C(D,T ),

and a faithful, exact, R-linear functor fT , such that:

1. T factorizes over DT−→ C(D,T )

fT−−→ R−Mod.

2. T satisfies the following universal property: Given

(a) another R-linear, abelian category A,

(b) an R-linear, faithful, exact functor, f : A → R−Mod,

(c) another representation F : D → A,

such that f F = T , then there exists a functor L(F ) - unique up to uniqueisomorphism of functors - such that the following diagram commutes:

C(D,T )

D R−Mod .

A

T

F f

fT

T

L(F )

The category C(D,T ) together with T and fT is uniquely determined by thisproperty up to unique equivalence of categories. It is explicitly described by thediagram category of Definition 6.1.10. It is functorial in D in the obvious sense.

The proof will be given in Section 6.4. We are going to view fT as an extensionof T from D to C(D,T ) and sometimes write simply T instead of fT .

The universal property generalizes easily.


Corollary 6.1.14. Let D, R, T be as in Theorem 6.1.19. Let A and f , F be asin loc.cit. 2. (a)-(c). Moreover, let S be a faithfully flat commutative unitaryR-algebra S and

φ : TS → (f F )S

an isomorphism of representations into S−Mod. Then there exists a functorL(F ) : C(D,T )→ A and an isomorphism of functors

φ : (fT )S → fS L(F )

such that

C(D,T )

D S−Mod

A

T

F fS

(fT )S

TS

L(F )

commutes up to φ and φ. The pair (L(F ), φ) is unique up to unique isomorphismof functors.

The proof will also be given in Section 6.4.

The following properties provide a better understanding of the nature of thecategory C(D,T ).

Proposition 6.1.15. 1. As an abelian category C(D,T ) is generated by theT v where v runs through the set of vertices of D, i.e., it agrees with itssmallest full subcategory such that the inclusion is exact containing allsuch T v.

2. Each object of C(D,T ) is a subquotient of a finite direct sum of objects ofthe form T v.

3. If α : v → v′ is an edge in D such that Tα is an isomorphism, then Tα isalso an isomorphism.

Proof. Let C′ ⊂ C(D,T ) be the subcategory generated by all T v. By definition,the representation T factors through C′. By the universal property of C(D,T ),we obtain a functor C(D,T ) → C′, hence an equivalence of subcategories ofR−Mod.


The second statement follows from the first criterion since the full subcategoryin C(D,T ) of subquotients of finite direct sums is abelian, hence agrees withC(D,T ). The assertion on morphisms follows since the functor fT : C(D,T )→R−Mod is faithful and exact between abelian categories. Kernel and cokernelof Tα vanish if kernel and cokernel of Tα vanish.

Remark 6.1.16. We will later give a direct proof, see Proposition 6.3.20. Itwill be used in the proof of the universal property.

The diagram category only weakly depends on T .

Corollary 6.1.17. Let D be a diagram and T, T ′ : D → R−Mod two represen-tations. Let S be a faithfully flat R-algebra and φ : TS → T ′S be an isomorphismof representations in S−Mod. Then it induces an equivalence of categories

Φ : C(D,T )→ C(D,T ′).

Proof. We apply the universal property of Corollary 6.1.14 to the representationT and the abelian category A = C(D,T ′). This yields a functor Φ : C(D,T )→C(D,T ′). By interchanging the role of T and T ′ we also get a functor Φ′ inthe opposite direction. We claim that they are inverse to each other. Thecomposition Φ′ Φ can be seen as the universal functor for the representationof D in the abelian category C(D,T ) via T . By the uniqueness part of theuniversal property, it is the identity.

Corollary 6.1.18. Let D2 be a diagram. Let T2 : D2 → R−Mod be a repre-sentation. Let

D2T2−→ C(D2, T2)

fT2−−→ R−Mod

be the factorization via the diagram category.

Let D1 ⊂ D2 be a full subdiagram. It has the representation T1 = T2|D1 obtainedby restricting T2. Let

D1T1−→ C(D1, T1)

fT1−−→ R−Mod

be the factorization via the diagram category. Let ι : C(D1, T1) → C(D2, T2) bethe functor induced from the inclusion of diagrams. Moreover, we assume thatthere is a representation F : D2 → C(D1, T1) compatible with T2, i.e., such thatthere is an isomorphism of functors

T2 → fT2 ι F = fT1 F .

Then ι is an equivalence of categories.

Proof. Let T ′2 = fT1 F : D2 → R−Mod and denote T ′1 = T ′2|D1

: D1 →R−Mod. Note that T2 and T ′2 and T1 and T ′1 are isomorphic by assumption.


By the universal property of the diagram category, the representation F inducesa functor

π′ : C(D2, T′2)→ C(D1, T1) .

It induces π : C(D2, T2) by precomposition with the equivalence Φ from Corol-lary 6.1.17. We claim that ι π and π ι are isomorphic to the identity functor.

By the uniqueness part of the universal property, the composition ι π′ :C(D2, T

′2) → C(D2, T2) is induced from the representation ι F of D2 in the

abelian category C(D2, T2). By the proof of Corollary 6.1.17 this is the equiva-lence Φ−1. In particular, ι π is the identity.

The argument for π ι on C(D1, T1) is analogous.

The most important ingredient for the proof of the universal property is thefollowing special case.

Theorem 6.1.19. Let R be a noetherian ring and A an abelian, R-linear cat-egory. Let

T : A −→ R−Mod

be a faithful, exact, R-linear functor and

A T−→ C(A, T )fT−→ R−Mod

the factorization via its diagram category (see Definition 6.1.10). Then T is anequivalence of categories.

The proof of this theorem will be given in Section 6.3.

6.1.4 Discussion of the Tannakian case

The above may be viewed as a generalization of Tannaka duality. We explainthis in more detail. We are not going to use the considerations in the sequel.

Let k be a field, C a k-linear abelian tensor category, and

T : C −→ k−Vect

a k-linear faithful tensor functor, all in the sense of [DM]. By standard Tan-nakian formalism (cf [Sa] and [DM]), there is a k-bialgebra A such that thecategory is equivalent to the category of A-comodules on finite dimensional k-vector spaces.

On the other hand, if we regard C as a diagram (with identities) and T as arepresentation into finite dimensional vector spaces, we can view the diagramcategory of C as the category A(C, T )−Comod by Theorem 6.1.12. By Theo-rem 6.1.19 the category C is equivalent to its diagram category A(C, T )−Comod.The construction of the two coalgebras A and A(C, T ) coincides. Thus Nori im-plicitely shows that we can recover the coalgebra structure of A just by lookingat the representations of C.


The algebra structure on A(C, T ) is induced from the tensor product on C (seealso Section 7.1). This defines a pro-algebraic scheme SpecA(C, T ). The coal-gebra structure turns SpecA(C, T ) into a monoid scheme. We may interpretA(C, T )−Comod as the category of finite-dimensional representations of thismonoid scheme.

If the tensor structure is rigid in addition, C(D,T ) becomes what Deligne andMilne call a neutral Tannakian category [DM]. The rigidity structure induces anantipodal map, making A(C, T ) into a Hopf algebra. This yields the structureof a group scheme on SpecA(C, T ), rather than only a monoid scheme.

We record the outcome of the discussion:

Theorem 6.1.20. Let R be a field and C be a neutral R-linear Tannakiancategory with faithful exact fibre functor T : C → R−Mod. Then A(C, T ) isequal to the Hopf algebra of the Tannakian dual.

Proof. By construction, see [DM] Theorem 2.11 and its proof.

A similar result holds in the case that R is a Dedekind domain and

T : D −→ R−Proj

a representation into finitely generated projective R-modules. Again by Theo-rem 6.1.12, the diagram category C(D,T ) equalsA(C, T )−Comod, whereA(C, T )is projective over R. Wedhorn shows in [Wed] that if SpecA(C, T ) is a groupscheme it is the same to have a representation of SpecA(C, T ) on a finitelygenerated R-module M and to endow M with an A(C, T )-comodule structure.

6.2 First properties of the diagram category

Let R be a unitary commutative noetherian ring, D a diagram and T : D →R−Mod a representation. We investigate the category C(D,T ) introduced inDefinition 6.1.10.

Lemma 6.2.1. If D is a finite diagram, then End(T ) is an R-algebra which isfinitely generated as an R-module.

Proof. For any p ∈ D the module Tp is finitely generated. Since R is noetherian,the algebra EndR(Tp) then is finitely generated as R-module. Thus End(T )becomes a unitary subalgebra of

∏p∈Ob(D) EndR(Tp). Since V (D) is finite and

R is noetherian,

End(T ) ⊂∏

p∈Ob(D)

EndR(Tp)

is finitely generated as R-module.

6.2. FIRST PROPERTIES OF THE DIAGRAM CATEGORY 137

Lemma 6.2.2. Let D be a finite diagram and T : D → R−Mod a representa-tion. Then:

1. Let S be a flat R-algebra. Then:

EndS(TS) = EndR(T )⊗ S

2. Let F : D′ → D be morphism of diagrams and T ′ = T F the inducedrepresentation. Then F induces a canonical R-algebra homomorphism

F ∗ : End(T )→ End(T ′) .

Proof. The algebra End(T ) is defined as a limit, i.e., a kernel

0→ End(T )→∏

p∈V (D)

EndR(Tp)φ−→

∏m∈D(p,q)

HomR(Tp, Tq)

with φ(p)(m) = eq Tm−Tm ep. As S is flat over R, this remains exact aftertensoring with S. As the R-module Tp is finitely presented and S flat, we have

EndR(Tp)⊗ S = EndS(TSp) .

Hence we get

0→ End(T |F )⊗ S →∏

p∈V (D)

EndS(TS(p))φ−→

∏m∈D(p,q)

HomS(TS(p), TS(q)) .

This is the defining sequence for End(TS).

The morphism of diagrams F : D′ → D induces a homomorphism∏p∈V (D)

EndR(Tp)→∏

p′∈V (D′)

EndR(T ′p′),

by mapping e = (ep)p to F ∗(e) with (F ∗(e))p′ = ef(p′) in EndR(T ′p′) =EndR(Tf(p′)). It is compatible with the induced homomorphism∏

m∈D(p,q)

HomR(Tp, Tq)→∏

m′∈D′(p′,q′)

HomR(T ′p′, T ′q′).

Hence it induces a homomorphism on the kernels.

Proposition 6.2.3. Let R be unitary commutative noetherian ring, D a finitediagram and T : D −→ R−Mod be a representation. For any p ∈ D the objectTp is a natural left End(T )-module. This induces a representation

T : D −→ End(T )−Mod,

such that T factorises via

DT−→ C(D,T )

fT−→ R−Mod.


Proof. For all p ∈ D the projection

pr : End(T )→ EndR(Tp)

induces a well-defined action of End(T ) on Tp making Tp into a left End(T )-module. To check that T is a representation of left End(T )-modules, we needTm ∈ HomR(Tp, Tq) to be End(T )-linear for all p, q ∈ D,m ∈ D(p, q). Thiscorresponds directly to the commutativity of the diagram in Remark 6.1.9.

Now let D be general. We study the system of finite subdiagrams F ⊂ D.Recall that subdiagrams are full, i.e., they have the same edges.

Corollary 6.2.4. The finite subdiagrams of D induce a directed system ofabelian categories

(C(D,T |F )

)F⊂Dfinite

with exact, faithful R-linear functors astransition maps.

Proof. The transition functors are induced from the inclusion via Lemma 6.2.2.

Recall that we have defined C(D,T ) as 2-colimit of this system, see Defini-tion 6.1.10.

Proposition 6.2.5. The 2-colimit C(D,T ) exists. It provides an R-linearabelian category such that the composition of the induced representation withthe forgetful functor

DT−→ C(D,T )

fT−−→ R−Modp 7→ Tp 7→ Tp.

yields a factorization of T . The functor fT is R-linear, faithful and exact.

Proof. It is a straightforward calculation that the limit category inherits allstructures of an R-linear abelian category. It has well-defined (co)products and(co)kernels because the transition functors are exact. It has a well-defined R-linear structure as all transition functors are R-linear. Finally, one shows thatevery kernel resp. cokernel is a monomorphism resp. epimorphism using thefact that all transition functors are faithful and exact.

So for every p ∈ D the R-module Tp becomes an End(T |F )-module for all finiteF ⊂ D with p ∈ F . Thus it represents an object in C(D,T ). This induces arepresentation

DT−→ C(D,T )

p 7→ Tp.

The forgetful functor is exact, faithful and R-linear. Composition with theforgetful functor fT obviously yields the initial diagram T .

We now consider functoriality in D.

6.2. FIRST PROPERTIES OF THE DIAGRAM CATEGORY 139

Lemma 6.2.6. Let D1, D2 be diagrams and G : D1 → D2 a map of diagrams.Let further T : D2 → R−Mod be a representation and

D2T−→ C(D2, T )

fT−−→ R−Mod

the factorization of T through the diagram category C(D2, T ) as constructed inProposition 6.2.5. Let

D1TG−−−→ C(D1, T G)

fTG−−−→ R−Mod

be the factorization of T G.

Then there exists a faithful R-linear, exact functor G, such that the followingdiagram commutes.

D1 D2

C(D1, T G) C(D2, T )

R−Mod

G

T G T

fTG fT

G

Proof. Let D1, D2 be finite diagrams first. Let T1 = T G|D1and T2 = T |D2

.The homomorphism

G∗ : End(T2)→ End(T1)

of Lemma 6.2.2 induces by restriction of scalars a functor on diagram categoriesas required.

Let now D1 be finite and D2 arbitrary. Let E2 be finite full subdiagram ofD2 containing G(D1). We apply the finite case to G : D1 → E2 and obtain afunctor

C(D1, T1)→ C(E2, T2)

which we compose with the canonical functor C(E2, T2)→ C(D2, T2). By func-toriality, it is independent of the choice of E2.

Let now D1 and D2 be arbitrary. For every finite subdiagram E1 ⊂ E1 we haveconstructed

C(E1, T1)→ C(D2, T2) .

They are compatible and hence define a functor on the limit.

Isomorphic representations have equivalent diagram categories. More precisely:


Lemma 6.2.7. Let T1, T2 : D → R−Mod be representations and φ : T1 → T2

an isomorphism of representations. Then φ induces an equivalence of categoriesΦ : C(D,T1)→ C(D,T2) together with an isomorphism of representations

φ : Φ T1 → T2

such that fT2 φ = φ.

Proof. We only sketch the argument which is analogous to the proof of Lemma 6.2.6.

It suffices to consider the case D = F finite. The functor

Φ : End(T1)−Mod→ End(T2)−Mod

is the extension of scalars for the R-algebra isomorphism End(T1) → End(T2)induced by conjugating by φ.

6.3 The diagram category of an abelian category

In this section we give the proof of Theorem 6.1.19: the diagram category of thediagram category of an abelian category with respect to a representation givenby an exact faithful functor is the abelian category itself.

We fix a commutative noetherian ring R with unit and an R-linear abelian cate-gory A. By R-algebra we mean a unital R-algebra, not necessarily commutative.

6.3.1 A calculus of tensors

We start with some general constructions of functors. We fix a unital R-algebraE, finitely generated as R-module, not necessarily commutative. The mostimportant case is E = R, but this is not enough for our application.

In the simpler case where R is a field, the constructions in this sections canalready be found in [DMOS].

Definition 6.3.1. Let E be an R-algebra which is finitely generated as R-module. We denote E−Mod the category of finitely generated left E-modules.

The algebra E and the objects of E−Mod are noetherian because R is.

Definition 6.3.2. Let A be an R-linear abelian category and p be an object ofA. Let E be a not necessarily commutative R algebra and

Eopf−→ EndA(p)

be a morphism of R-algebras. We say that p is a right E-module in A.

6.3. THE DIAGRAM CATEGORY OF AN ABELIAN CATEGORY 141

Example 6.3.3. Let A be the category of left R′-modules for some R-algebraR′. Then a right E-module in A is the same thing as an (R′, E)-bimodule, i.e.,a left R′-module with the structure of a right E-module.

Lemma 6.3.4. Let A be an R-linear abelian category and p be an object of A.Let E be a not necessarily commutative R-algebra and p a right E-module in A.Then

HomA(p, ) : A → R−Mod

can naturally be viewed as a functor to E−Mod.

Proof. For every q ∈ A, the algebra E operates on HomA(p, q) via functoriality.

Proposition 6.3.5. Let A be an R-linear abelian category and p be an objectof A. Let E be a not necessarily commutative R algebra and p a right E-modulein A. Then the functor

HomA(p, ) : A −→ E−Mod

has an R-linear left adjoint

p⊗E : E−Mod −→ A.

It is right exact. It satisfiesp⊗E E = p,

and on endomorphisms of E we have (using EndE(E) ∼= Eop)

p⊗E : EndE(E) −→ EndA(p)a 7−→ f(a).

Proof. Right exactness of p⊗E follows from the universal property. For everyM ∈ E−Mod, the value of p ⊗E M is uniquely determined by the universalproperty. In the case of M = E, it is satisfied by p itself because we have forall q ∈ A

HomA(p, q) = HomE(E,HomA(p, q)).

This identification also implies the formula on endomorphisms of M = E.

By compatibility with direct sums, this implies that p ⊗E En =⊕n

i=1 p for

free E-modules. For the same reason, morphisms Em(aij)ij−−−−→ En between free

E-modules must be mapped to⊕m

i=1 pf(aij)ij−−−−−→

⊕ni=1 p.

Let M be a finitely presented left E-module. We fix a finite presentation

Em1(aij)ij−−−−→ Em0

πaM → 0.

Since p⊗E preserves cokernels (if it exists), we need to define

p⊗E M := Coker(pm1A:=f(aij)ij−−−−−−−→ pm0).


We check that it satisfies the universal property. Indeed, for all q ∈ A, we havea commutative diagram

HomA(p⊗ Em1 , q)

∼=

HomA(p⊗ Em0 , q)oo

∼=

HomA(p⊗M, q)oo

0oo

HomE(Em1 ,HomA(p, q)) HomE(Em0 ,HomA(p, q))oo HomE(M,HomA(p, q))oo 0oo

Hence the dashed arrow exists and is an isomorphism.

The universal property implies that p ⊗E M is independent of the choice ofpresentation and functorial. We can also make this explicit. For a morphismbetween arbitrary modules ϕ : M → N we choose lifts

Em1 Em0 M 0

En1 En0 N 0.

A πA

B πB

ϕ1 ϕ0 ϕ

The respective diagram in A,

pm1 pm0 Coker(A) 0

pn1 pn0 Coker(B) 0.

A πA

B πB

ϕ1 ϕ0 ∃!

induces a unique morphism p ⊗E (ϕ) : p ⊗E M → p ⊗E N that keeps thediagram commutative. It is independent of the choice of lifts as different lifts ofprojective resolutions are homotopic. This finishes the construction.

Corollary 6.3.6. Let E be an R-algebra finitely generated as R-module and Aan R-linear abelian category. Let

T : A −→ E−Mod

be an exact, R-linear functor into the category of finitely generated E-modules.Further, let p be a right E-module in A with structure given by

Eopf−→ EndA(p)

a morphism of R-algebras. Then the composition

Eopf−→ EndA(p)

T−→ EndE(Tp).


induces a right action on Tp, making it into an E-bimodule. The composition

E−Modp⊗E−→ A T−→ E−Mod

M 7→ p⊗E M 7→ Tp⊗E M

becomes the usual tensor functor of E-modules.

Proof. It is obvious that the composition

E−Modp⊗E−→ A T−→ E−Mod

En 7→ p⊗E En 7→ Tp⊗E En

induces the usual tensor functor

Tp⊗E : E−Mod −→ E−Mod

on free E-modules. For arbitrary finitely generated E-modules this follows fromthe fact that Tp⊗E is right exact and T is exact.

Remark 6.3.7. Let E be an R-algebra, let M be a right E-module and N bea left E-module. We obtain the tensor product M ⊗E N by dividing out theequivalence relation m · e ⊗ n ∼ m ⊗ e · n for all m ∈ M,n ∈ N, e ∈ E of thetensor product M ⊗RN of R-modules. We will now see that a similar approachholds for the abstract tensor products p⊗RM and p⊗E M in A as defined inProposition 6.3.5. For the easier case that R is a field, this approach has beenused in [DM].

Lemma 6.3.8. Let A be an R-linear, abelian category, E a not necessarilycommutative R-algebra which is finitely generated as R-module and p ∈ A aright E-module in A. Let M ∈ E−Mod and E′ ∈ E−Mod be in addition aright E-module. Then p⊗E E′ is a right E-module in A and we have

p⊗E (E′ ⊗E M) = (p⊗E E′)⊗E M.

Moreover,

(p⊗E E)⊗RM = p⊗RM.

Proof. The right E-module structure on p⊗EE′ is defined by functoriality. Theequalities are immediate from the universal property.

Proposition 6.3.9. Let A be an R-linear, abelian category. Let further E be aunital R-algebra with finite generating family e1, . . . , em. Let p a right E-modulein A with structure given by

Eopf−→ EndA(p).

Let M be a left E-module.


Then p⊗E M is isomorphic to the cokernel of the map

Σ :

m⊕i=1

(p⊗RM) −→ p⊗RM

given bym∑i=1

(f(ei)⊗ idM − idp ⊗ eiidM )πi

with πi the projection to the i-summand.

More suggestively (even if not quite correct), we write

Σ : (xi ⊗ vi)mi=1 7→m∑i=1

(f(ei)(xi)⊗ vi − xi ⊗ (ei · vi))

for xi ∈ p and vi ∈M .

Proof. Consider the sequence

m⊕i=1

E ⊗R E −→ E ⊗ E −→ E −→ 0

where the first map is given by

(xi ⊗ yi)mi=1 7→m∑i=1

xiei ⊗ yi − xi ⊗ eiyi

and the second is multiplication. We claim that it is exact. The sequenceis exact in E because E is unital. The composition of the two maps is zero,hence the cokernel maps to E. The elements in the cokernel satisfy the relationxei ⊗ y = x ⊗ eiy for all x, y and i = 1, . . . ,m. The ei generate E, hencexe ⊗ y = x ⊗ ey for all x, y and all e ∈ E. Hence the cokernel equals E ⊗E Ewhich is E via the multiplication map.

Now we tensor the sequence from the left by p and from the right by M andobtain an exact sequence

m⊕i=1

p⊗E (E ⊗R E)⊗E M −→ p⊗E (E ⊗R E)⊗E M −→ p⊗E E ⊗E M → 0.

Applying the computation rules of Lemma 6.3.8, we get the sequence in theproposition.

Similarly to Proposition 6.3.5 and Corollary 6.3.6, but less general, we constructa contravariant functor HomR(p, ) :


Proposition 6.3.10. Let A be an R-linear abelian category, and p be an objectof A. Then the functor

HomA( , p) : A −→ R−Mod

has a left adjointHomR( , p) : R−Mod −→ A.

This means that for all M ∈ R−Mod and q ∈ A, we have

HomA(q,HomR(M,p)) = HomR(M,HomA(q, p)).

It is left exact. It satisfiesHomR(R, p) = p.

IfT : A −→ R−Mod

is an exact, R-linear functor into the category of finitely generated R-modulesthen the composition

R−ModHom( ,p)−→ A T−→ R−Mod

M 7→ HomR(M,p) 7→ HomR(M,Tp)

is the usual Hom( , Tp)-functor in R−Mod.

Proof. The arguments are the same as in the proof of Proposition 6.3.5 andCorollary 6.3.6.

Remark 6.3.11. LetA be anR-linear, abelian category. The functors HomR( , p)as defined in Proposition 6.3.10 and p ⊗R as defined in Proposition 6.3.6 arealso functorial in p, i.e., we have even functors

HomR( , ) : (R−Mod) ×A −→ A

and⊗R : A×R−Mod −→ A.

We will denote the image of a morphism pα−→ q under the functor HomR(M, )

byHomR(M,p)

α−→ HomR(M, q)

This notation α is natural since by composition

A Hom(M, )−→ A T−→ R−Modp 7→ HomR(M,p) 7→ HomR(M,Tp)

T (α) becomes the usual left action of Tα on HomR(M,Tp).


Proof. This follows from the universal property.

We will now check that the above functors have very similar properties to usualtensor and Hom-functors in R−Mod.

Lemma 6.3.12. Let A be an R-linear, abelian category and M a finitely gen-erated R-module. Then the functor HomR(M, ) is right-adjoint to the functor⊗RM .

If

T : A −→ R−Mod

is an R-linear, exact functor into finitely generated R-modules, the composedfunctors T HomR(M, ) and T ( ⊗RM) yield the usual hom-tensor adjunctionin R−Mod.

Proof. The assertion follows from the universal property and the identificationT HomR(M, ) = HomR(M,T ) in Proposition 6.3.10 and T ⊗R M =(T )⊗RM in Proposition 6.3.6.

6.3.2 Construction of the equivalence

Definition 6.3.13. Let A be an abelian category and S a not necessarilyabelian subcategory. With 〈S〉 we denote the smallest full abelian subcate-gory of A containing S, i.e., the intersection of all full subcategories of A thatare abelian, contain S, and for which the inclusion functor is exact.

Lemma 6.3.14. Let A = 〈F 〉 for a finite set of objects. Let T : 〈F 〉 → R−Modbe a faithful exact functor. Then the inclusion F → 〈F 〉 induces an equivalence

End(T |F )−Mod −→ C(〈F 〉, T ).

Proof. Let E = End(T |F ). Its elements are tuples of endomorphisms of Tp forp ∈ F commuting with all morphisms p→ q in F .

We have to show that E = End(T ). In other words, that any element of Edefines a unique endomorphism of Tq for all objects q of 〈F 〉 and commuteswith all morphisms in 〈F 〉.Any object q is a subquotient of a finite direct sum of copies of objects p ∈ F .The operation of E on Tp with p ∈ F extends uniquely to an operation on directsums, kernels and cokernels of morphisms. It is also easy to see that the actioncommutes with Tf for all morphisms f between these objects. This means thatit extends to all objects 〈F 〉, compatible with all morphisms.

We first concentrate on the caseA = 〈p〉. From now on, we abbreviate End(T |p)by E(p).


Lemma 6.3.15. Let 〈p〉 = A be an abelian category. Let 〈p〉 T−→ R−Mod afaithful exact R-linear functor into the category of finitely generated R-modules.Let

〈p〉 T−→ E(p)−ModfT−→ R−Mod

be the factorization via the diagram category of T constructed in Proposition 6.2.5.Then:

1. There exists an object X(p) ∈ Ob(〈p〉) such that

T (X(p)) = E(p).

2. The object X(p) has a right E(p)-module structure in A

E(p)op → EndA(X(p))

such that the induced E(p)-module structure on E(p) is the product.

3. There is an isomorphism

τ : X(p)⊗E(p) T p→ p

which is natural in f ∈ EndA(p), i.e.,

p p

X(p)⊗E(p) T p X(p)⊗E(p) T p

f

id⊗ T f

τ τ

An easier construction of X(p) in the field case can be found in [DM], theconstruction for R being a noetherian ring is due to Nori [N].

Proof. We consider the object HomR(Tp, p) ∈ A. Via the contravariant functor

R−ModHom( ,p)−→ A

Tp 7→ HomR(Tp, p)

of Proposition 6.3.10 it is a right EndR(Tp)-module in A which, after apply-ing T just becomes the usual right End(Tp)-module HomR(Tp, Tp). For eachϕ ∈ End(Tp), k we will write ϕ for the action on Hom(Tp, p) as well. ByLemma 6.3.12 the functors HomR(Tp, ) and ⊗R Tp are adjoint, so we obtainan evaluation map

ev : HomR(Tp, p)⊗R Tp −→ p


that becomes the usual evaluation in R−Mod after applying T . Our aim is nowto define X(p) as a suitable subobject of HomR(Tp, p) ∈ A. The structures onX(p) will be induced from the structures on HomR(Tp, p).

Let M ∈ R−Mod. We consider the functor

A HomR(M, )−→ Ap 7→ HomR(M,p)

of Remark 6.3.11. The endomorphism ring EndA(p)) ⊂ EndR(Tp) is finitelygenerated as R-module, since T is faithful and R is noetherian. Let α1, ..., αnbe a generating family. Since

E(p) = ϕ ∈ End(Tp)|Tα ϕ = ϕ Tα ∀α : p→ p,

we can write E(p) as the kernel of

Hom(Tp, Tp) −→⊕n

i=1 Hom(Tp, Tp)u 7→ u Tαi − Tαi u

By the exactness of T , the kernel X(p) of

Hom(Tp, p) −→⊕n

i=1 Hom(Tp, p)u 7→ u Tαi − αi u

is a preimage of E(p) under T in A.

By construction, the right EndR(Tp)-module structure on HomR(Tp, p) restrictsto a right E(p)-module structure on X(p) whose image under T yields thenatural E(p) right-module structure on E(p).

Now consider the evaluation map

ev : HomR(Tp, p)⊗R Tp −→ p

mentioned at the beginning of the proof. By Proposition 6.3.9 we know thatthe cokernel of the map Σ defined there is isomorphic to X(p) ⊗E(p) T p. Thediagram

⊕ki=1(X(p)⊗R Tp) X(p)⊗R Tp HomR(Tp, p)⊗R Tp p

X(p)⊗E(p) T p

Σ incl⊗ id ev

Coker(Σ)

in A maps via T to the diagram


⊕ki=1(E(p)⊗R Tp) E(p)⊗R Tp HomR(Tp, Tp)⊗R Tp Tp

E(p)⊗E(p) T p

Σ incl⊗ id ev

Coker(Σ)

in R−Mod, where the composition of the horizontal maps becomes zero. SinceT is faithful, the respective horizontal maps in A are zero as well and induce amap

τ : X(p)⊗E(p) Tp −→ p

that keeps the diagram commutative. By definition of Σ in Proposition 6.3.9,the respective map

T τ : E(p)⊗E(p) T p −→ T p

becomes the natural evaluation isomorphism of E-modules. Since T is faithful,τ is an isomorphism as well.

Naturality in f holds since T is faithful and

T p T p

E(p)⊗E(p) T p E(p)⊗E(p) T p

T f

id⊗ T f

T τ T τ

commutes in E(p)−Mod.

Proposition 6.3.16. Let 〈p〉 = A be an R-linear, abelian category and

A T−→ R−Mod

be as in Theorem 6.1.19. Let

A T−→ C(A, T )fT−→ R−Mod

be the factorization of T via its diagram category. Then T is an equivalence ofcategories with inverse given by X(p) ⊗E(p) with X(p) the object constructedin Lemma 6.3.15.

Proof. We have A = 〈p〉, thus C(A, T ) = E(p)−Mod. Consider the object X(p)of Lemma 6.3.15. It is a right E(p)-module in A, in other words

f :(E(p)

)op −→ EndA(X(p))ϕ 7−→ ϕ


We apply Corollary 6.3.6 to E = E(p), the object X(p), the above f and thefunctor

T : 〈p〉 −→ E(p)−Mod .

It yields the functor

X(p)⊗E(p) : E(p)−Mod −→ 〈p〉

such that the composition

E(p)−ModX(p)⊗E(p)−→ 〈p〉 T−→ E(p)−Mod

M 7−→ X(p)⊗E(p) M 7→ T (X(p))⊗E(p) M = E(p)⊗E(p) M

becomes the usual tensor product of E(p)-modules, and therefore yields theidentity functor.

We want to check that X(p)⊗E(p) is a left-inverse functor of T as well. Thuswe need to find a natural isomorphism τ , i.e., for all objects p1, p2 ∈ A weneed isomorphisms τp1

, τp2such that for morphisms f : p1 → p2 the following

diagram commutes:

X(p)⊗E(p) T p1 X(p)⊗E(p) T p2

p1 p2

id⊗ T f

f

τp1 τp2

Since the functors T and fT are faithful and exact, and we have T = ft T ,we know that T is faithful and exact as well. We have already shown thatT X(p)⊗E(p) is the identity functor. So X(p)⊗E(p) is faithful exact as well.Since A is generated by p, it suffices to find a natural isomorphism for p and itsendomorphisms. This is provided by the isomorphism τ of Lemma 6.3.15.

Proof of Theorem 6.1.19. If A is generated by one object p, then the functor Tis an equivalence of categories by Proposition 6.3.16. It remains to reduce tothis case.

The diagram category C(A, T ) arises as a direct limit, hence we have

2−colimF⊂Ob(A)End(T |F )−Mod

and in the same way we have

A = 2−colimF⊂Ob(A)〈F 〉


with F ranging over the system of full subcategories of A that contain only afinite number of objects. Moreover, by Lemma 6.3.14, we have End(T |F ) =End(T |〈F 〉). Hence it suffices to check equivalence of categories

〈F 〉T |〈F〉−→ End(T |F )−Mod

for all abelian categories that are generated by a finite number of objects.

We now claim that 〈F 〉 ∼= 〈⊕

p∈F p〉 are equivalent: indeed, since any endomor-phism of

⊕p∈F p is of the form (apq)p,q∈F with apq : p → q, and since F has

all finite direct sums, we know that 〈⊕

p∈F p〉 is a full subcategory of 〈F 〉. Onthe other hand, for any q, q′ ∈ F the inclusion q →

⊕p∈F p is a kernel and the

projection⊕

p∈F p q′ is a cokernel, so for any set of morphisms (aqq′)q,q′∈F ,the morphism aqq′ : q → q′ by composition

q →⊕p∈F

(app′ )p,p′∈F−−−−−−−−→⊕p′∈F

p′ q′

is contained in 〈⊕

p∈F p〉. Thus 〈F 〉 is a full subcategory of 〈⊕

p∈F p〉.Similarly one sees that End(T |p)−Mod is equivalent to End(T |F )−Mod. Sowe can even assume that our abelian category is generated by just one objectq =

⊕p∈F p.

6.3.3 Examples and applications

We work out a couple of explicit examples in order to demonstrate the strengthof Theorem 6.1.19. We also use the arguments of the proof to deduce an ad-ditional property of the diagram property as a first step towards its universalproperty.

Throughout let R be a noetherian unital ring.

Example 6.3.17. Let T : R−Mod → R−Mod be the identity functor viewedas a representation. Note that R−Mod is generated by the object Rn. ByTheorem 6.1.19 and Lemma 6.3.14, we have equivalences of categories

End(T |Rn)−Mod −→ C(R−Mod, T ) −→ R−Mod.

By definition, E = End(T |Rn) consists of those elements of EndR(Rn) whichcommute with all elements of EndA(Rn), i.e., the center of the matrix algebra,which is R.

This can be made more interesting by playing with the representation.

Example 6.3.18 (Morita equivalence). Let R be a noetherian commutativeunital ring, A = R−Mod. Let P be a flat finitely generated R-module and

T : R−Mod −→ R−Mod, M 7→M ⊗R P.


It is faithful and exact, hence the assumptions of Theorem 6.1.19 are satisfiedand we get an equivalence

C(R−Mod, T ) −→ R−Mod .

Note that A = 〈R〉 and hence by Lemma 6.3.14, C(R−Mod, T ) = E−Mod withE = EndR(T |R) = EndR(P ). Hence we have shown that

EndR(P )−Mod→ R−Mod

is an equivalence of categories. This is a case of Morita equivalence of categoriesof modules.

Example 6.3.19. Let R be a noetherian commutative unital ring and E anR-algebra finitely generated as an R-module. Let

T : E−Mod→ R−Mod

be the forgetful functor. The category E−Mod is generated by the module E.Hence by Theorem 6.1.19 and Lemma 6.3.14, we have again equivalences ofcategories

E′−Mod −→ C(E−Mod, T ) −→ E−Mod,

where E′ = End(T |E) is the subalgebra of EndR(E) of endomorphisms com-patible with all E-morphisms E → E. Note that EndE(E) = Eop and hence E′

is the centralizer of Eop in EndR(E)

E′ = CEndR(E)(Eop) = E .

Hence in this case the functor A → C(A, T ) is the identity.

We deduce another consequence of the explicit description of C(D,T ).

Proposition 6.3.20. Let D be a diagram and T : D → R−Mod a representa-tion. Let

DT−→ C(D,T )

fT−−→ R−Mod

its factorization. Then the category C(D,T ) is generated by the image of T :

C(D,T ) = 〈T (D)〉 .

Proof. It suffices to consider the case when D is finite. Let X =⊕

p∈D Tp andE = EndR(X). Let S ⊂ E be the R-subalgebra generated by Te for e ∈ E(D)and the projectors pp : X → T (p). Then

E = End(T ) = CE(S)

is the commutator of S in E. (The endomorphisms commuting with the projec-tors are those respecting the decomposition. By definition, End(T ) consists ofthose endomorphisms of the summands commuting with all Te.)

6.4. UNIVERSAL PROPERTY OF THE DIAGRAM CATEGORY 153

By construction C(D,T ) = E−Mod. We claim that it is equal to

A := 〈T p|p ∈ D〉 = 〈X〉

with X =⊕

p∈D T p. The category has a faithful exact representation by

fT |A. Note that fT (X) = X. By Theorem 6.1.19, the category A is equiv-alent to its diagram category C(〈X〉, fT ) = E′−Mod with E′ = End(fT |A).By Lemma 6.3.14, E′ consists of elements of E commuting with all elements ofEndA(X). Note that

EndA(X) = EndE(X) = CE(E)

and henceE′ = CE(CE(E)) = CE(CE(CE(S)) = CE(S)

because a triple commutator equals the simple commutator. We have shownE = E′ and the two categories are equivalent.

Remark 6.3.21. This is a direct proof of Proposition 6.1.15.

6.4 Universal property of the diagram category

At the end of this section we will be able to establish the universal property ofthe diagram category.

Let T : D −→ R−Mod be a diagram and

DT−→ C(D,T )

fT−−→ R−Mod

the factorization of T via its diagram category. Let A be another R-linearabelian category, F : D → A a representation, and f : A → R−Mod a faithful,exact, R-linear functor into the categories of finitely generated R-modules suchthat f F = T .

Our aim is to deduce that there exists - uniquely up to isomorphism - an R-linearexact faithful functor

L(F ) : C(D,T )→ A,making the following diagram commute:

D

C(D,T ) A

R−Mod

T

F

fT

TA

∃!L(F )


Proposition 6.4.1. There is a functor L(F ) making the diagram commute.

Proof. We can regard A as a diagram and obtain a representation

A TA−−→ R−Mod,

that factorizes via its diagram category

A TA−−→ C(A, TA)fTA−−→ R−Mod.

We obtain the following commutative diagram

D A

C(D,T ) C(A, TA)

R−Mod

TD

F

fT

TA

fTA

T TA

By functoriality of the diagram category (see Proposition 6.2.6) there exists anR-linear faithful exact functor F such that the following diagram commutes:

D A

C(D,T ) C(A, TA)

R−Mod

TD

F

fT

TA

fTA

F

Since A is R-linear, abelian, and T is faithful, exact, R-linear, we know byProposition 6.1.19, that TA is an equivalence of categories. The functor

L(F ) : C(D,T )→ A,

is given by the composition of F with the inverse of TA. Since an equivalenceof R-linear categories is exact, faithful and R-linear, L(F ) is so as well, as it isthe composition of such functors.

Proposition 6.4.2. The functor L(F ) is unique up to unique isomorphism.

6.4. UNIVERSAL PROPERTY OF THE DIAGRAM CATEGORY 155

Proof. Let L′ be another functor satisfying the condition in the diagram. Let C′be the subcategory of C(D,T ) on which L′ = L(F ). We claim that the inclusionis an equivalence of categories. Without loss of generality, we may assume D isfinite.

Note that the subcategory is full because f : A → R−Mod is faithful. It containsall objects of the form T p for p ∈ D. As the functors are additive, this impliesthat they also have to agree (up to canonical isomorphism) on finite direct sumsof objects. As the functors are exact, they also have to agree on and all kernelsand cokernels. Hence C′ is the full abelian subcategory of C(D,T ) generated byT (D). By Proposition 6.3.20 this is all of C(D,T ).

Proof of Theorem 6.1.13. Let T : D → R−Mod be a representation and f :A → R−Mod, F : D → A as in the statement. By Proposition 6.4.1 thefunctor L(F ) exists. It is unique by Proposition 6.4.2. Hence C(D,T ) satisfiesthe universal property of Theorem 6.1.13.

Let C be another category satisfying the universal property. By the universalproperty for C(D,T ) and the representation of D in C, we get a functor Ψ :C(D,T ) → C. By interchanging their roles, we obtain a functor Ψ′ in theopposite direction. Their composition Ψ′ Ψ satisfies the universal property forC(D,T ) and the representation T . By the uniqueness part, it is isomorphic tothe identity functor. The same argument also applies to Ψ Ψ′. Hence Ψ is anequivalence of categories.

Functoriality of C(D,T ) in D is Lemma 6.2.6.

The generalized universal property follows by a trick.

Proof of Corollary 6.1.14. Let T : D → R−Mod, f : A → R−Mod und F :D → A be as in the corollary. Let S be a faithfully flat R-algebra and

φ : TS → (f F )S

an isomorphism of representations into S−Mod. We first show the existence ofL(F ).

Let A′ be the category with objects of the form (V1, V2, ψ) where V1 ∈ R−Mod,V2 ∈ A and ψ : V1⊗R S → f(V2)⊗R S an isomorphism. Morphisms are definedas pairs of morphisms in R−Mod and A such the obvious diagram commutes.This category is abelian because S is flat over R. Kernels and cokernels aretaken componentwise. Let f ′ : A′ → R−Mod be the projection to the firstcomponent. It is faithful and exact because S is faithfully flat over R.

The data T , F and φ define a representation F ′ : D → A′ compatible with T .By the universal property of Theorem 6.1.13, we obtain a factorization

F ′ : DT−→ C(D,T )

L(F ′)−−−−→ A′ .


We define L(F ) as the composition of L(F ′) with the projection to the secondcomponent. The transformation

φ : (fT )S → fS L(F )

is defined onX ∈ C(D,T ) using the isomorphism ψ part of the object L(F ′)(X) ∈A′.Conversely, the triple (f, L(F ), φ) satisfies the universal property of L(F ′). Bythe uniqueness part of the universal property, this means that it agrees withL(F ′). This makes L(F ) unique.

6.5 The diagram category as a category of co-modules

Under more restrictive assumptions on R and T , we can give a description ofthe diagram category of comodules, see Theorem 6.1.12.

6.5.1 Preliminary discussion

In [DM] Deligne and Milne note that if R is a field, E a finite-dimensional R-algebra, and V an E-module that is finite-dimensional as R-vector space thenV has a natural structure as comodule over the coalgebra E∨ := HomR(E,R).For an algebra E finitely generated as an R-module over an arbitrary noetherianring R, the R-dual E∨ does not even necessarily carry a natural structure of anR-coalgebra. The problem is that the dual map to the algebra multiplication

E∨µ∗−→ (E ⊗R E)∨

does not generally define a comultiplication because the canonical map

ρ : E∨ ⊗R E∨ → Hom(E,E∨) ∼= (E ⊗R E)∨

fails to be an isomorphism in general. In this chapter we will see that thisisomorphism holds true for the R-algebras End(TF ) if we assume that R is aDedekind domain or field. We will then show that by

C(D,T ) = 2−colimF⊂D(End(TF )−Mod)

= 2−colimF⊂D(End(TF )∨−Comod) = (2−colimF⊂DEnd(TF )∨)−Comod

we can view the diagram category C(D,T ) as the category of finitely generatedcomodules over the coalgebra 2−colimF⊂DEnd(TF )∨.

Remark 6.5.1. Note that the category of comodules over an arbitrary coal-gebra C is not abelian in general, since the tensor product X ⊗R − is rightexact, but in general not left exact. If C is flat as R-algebra (e.g. free), thenthe category of C-comodules is abelian [MM, pg. 219].

6.5. THE DIAGRAM CATEGORY AS A CATEGORY OF COMODULES157

6.5.2 Coalgebras and comodules

Let R be a noetherian ring with unit.

Proposition 6.5.2. Let E be an R-algebra which is finitely generated as R-module. Then the canonical map

ρ : E∨ ⊗RM → Hom(E,M)ϕ⊗m 7→ (n 7→ ϕ(n) ·m)

becomes an isomorphism for all R-modules M if and only if E is projective.

Proof. [Str, Proposition 5.2]

Lemma 6.5.3. Let E be an R-algebra which is finitely generated and projectiveas an R-module.

1. The R-dual module E∨ carries a natural structure of a counital coalgebra.

2. Any left E-module that is finitely generated as R-module carries a naturalstructure as left E∨-comodule.

3. We obtain an equivalence of categories between the category of finitelygenerated left E-modules and the category of finitely generated left E∨-comodules.

Proof. By the repeated application of Proposition 6.5.2, this becomes a straight-forward calculation. We will sketch the main steps of the proof.

1. If we dualize the associativity constraint of E we obtain a commutativediagram of the form

(E ⊗R E ⊗R E)∨ (E ⊗R E)∨

(E ⊗R E)∨ E∨.

(µ⊗ id)∗

(id⊗ µ)∗

µ∗

µ∗

By the use of the isomorphism in Propostion 6.5.2 and Hom-Tensor ad-junction we obtain the commutative diagram

E∨ ⊗R E∨ ⊗R E∨ E∨ ⊗R E∨

E∨ ⊗R E∨ E∨,

µ∗ ⊗ id∗

id∗ ⊗ µ∗

µ∗

µ∗


which induces a cocommutative comultiplication on E∨. Similarly we ob-tain the counit diagram, so E∨ naturally gets a coalgebra structure.

2. For an E-module M we analogously dualize the respective diagram

M E ⊗RM

E ⊗RM E ⊗R E ⊗RM

m

m id⊗m

µ⊗ id

and use Proposition 6.5.2 and Hom-Tensor adjunction to see that the E-multiplication induces a well-defined E∨-comultiplication

M E∨ ⊗RM

E∨ ⊗RM E∨ ⊗R E∨ ⊗RM

m

m

µ∗ ⊗ id

id⊗ m

on M .

3. For any homomorphism f : M −→ N of left E-modules, the commutativediagram

M N

E ⊗RM E ⊗R N

f

id⊗ f

µ µ

induces a commutative diagram

E∨ ⊗RM E∨ ⊗R N,

M N

id⊗ f

f

µ µ

thus f is a homomorphism of left E∨-comodules.

6.5. THE DIAGRAM CATEGORY AS A CATEGORY OF COMODULES159

4. Conversely, we can dualize the E∨-comodule structure to obtain a (E∨)∨ =E-module structure. The two constructions are inverse to each other.

Definition 6.5.4. Let A be a coalgebra over R. Then we denote by A−Comodthe category of comodules over A that are finitely generated as a R-modules.

Recall that R−Proj denotes the category of finitely generated projective R-modules.

Corollary 6.5.5. Let R be a field or Dedekind domain, D a diagram and

T : D −→ R−Proj

a representation. Set A(D,T ) := lim−→F⊂D finiteEnd(TF )∨. Then A(D,T ) has

the structure of a coalgebra and the diagram category of T is the abelian categoryA(D,T )−Comod.

Proof. For any finite subset F ⊂ D the algebra End(TF ) is a submodule ofthe finitely generated projective R-module

∏p∈F End(Tp). Since R is a field or

Dedekind domain, for a finitely generated module to be projective is equivalentto being torsion free. Hence the submodule End(TF ) is also finitely generatedand torsion-free, or equivalently, projective. By the previous lemma, End(TF )∨

is an R-coalgebra and End(TF )−Mod ∼= End(TF )∨−Comod. From now on, wedenote End(TF )∨ with A(F, T ). Taking limits over the direct system of finitesubdiagrams as in Definition 6.1.10, we obtain

C(D,T ) := 2−colimF⊂D finiteEnd(TF )−Mod

= 2−colimF⊂D finiteA(F, T )−Comod.

Since the category of coalgebras is cocomplete, A(D,T ) = lim−→F⊂D A(F, T ) is a

coalgebra as well.

We now need to show that the categories 2−colimF⊂D finite(A(F, T )−Comod)andA(D,T )−Comod are equivalent. For any finite F the canonical mapA(F, T ) −→A(D,T ) via restriction of scalars induces a functor

φF : A(F, T )−Comod −→ A(D,T )−Comod

and therefore by the universal property a unique functor

u : lim−→A(F, T )−Comod −→ A(D,T )−Comod.

such that for all finite F ′, F ′′ ⊂ D with F ′ ⊂ F ′′ and the canonical functors

ψF : A(F ′, T )−Comod −→ lim−→F⊂D

A(F, T )−Comod

the following diagram commutes:


A(F ′, T )−Comod A(F ′′, T )−Comod

lim−→F⊂D(A(F, T )−Comod)

A(D,T )−Comod

φF ′F ′′

ψF ′

φF ′

ψF ′′

φF ′′

∃!u

We construct an inverse map to u: Let M be an A(D,T )-comodule and

m : M →M ⊗R A(D,T )

be the comultiplication. Let M = 〈x1, .., xn〉R. Then m(xi) =∑nk=1 aki ⊗ xk

for certain aki ∈ A(D,T ). Every aki is already contained in an A(F, T ) forsufficiently large F . By taking the union of these finitely many F , we canassume that all aki are contained in one coalgebra A(F, T ). Since x1, .., xngenerate M as R-module, m defines a comultiplication

m : M →M ⊗R A(F, T ).

So M is an A(F, T )-comodule in a natural way, thus via ψF an object of2−colimI(Ai−Comod).

We also need to understand the behavior of A(D,T ) under base-change.

Lemma 6.5.6 (Base change). Let R be a field or a Dedekind domain andT : D → R−Proj a representation. Let R→ S be flat. Then

A(D,TS) = A(D,T )⊗R S .

Proof. Let F ⊂ D be a finite subdiagram. Recall that

A(F, T ) = HomR(End(T |F ), R) .

Both R and EndR(T |F ) are projective because R is a field or a Dedekind domain.Hence by Lemma 6.2.2

HomR(EndR(T |F ), R)⊗S ∼= HomS(EndR(T |F )⊗S, S) ∼= HomS(EndS((TS)|F ), S).

This is nothing but A(F, TS). Tensor products commute with direct limits,hence the statement for A(D,T ) follows immediately.

Chapter 7

More on diagrams

We study additional structures on a diagram and a representation that lead tothe construction of a tensor product on the diagram category. The aim is thento turn it into a rigid tensor category with a faithful exact functor to a categoryof R-modules. The chapter is formal, but the assumptions are tailored to theapplication to Nori motives.

A particularly puzzling and subtle question is how the question of graded com-mutativity of the Kunneth formula is dealt with.

We continue to work in the setting of Chapter 6.

7.1 Multiplicative structure

Let R a fixed noetherian unital commutative ring.

Recall that R−Proj is the category of projective R-modules of finite type overR. We only consider representations T : D −→ R−Proj where D is a diagramwith identities, see Definition 6.1.1.

Definition 7.1.1. Let D1, D2 be diagrams with identities. Then D1 × D2 isdefined as the diagram with vertices of the form (v, w) for v a vertex of D1, wa vertex of D2, and with edges of the form (α, id) and (id, β) for α an edge ofD1 and β an edge of D2 and with id = (id, id).

Remark 7.1.2. Levine in [L1] p.466 seems to define D1 × D2 by taking theproduct of the graphs in the ordinary sense. He claims (in the notation of loc.cit.) a map of diagrams

H∗Sch′ ×H∗Sch′ → H∗Sch′.

It is not clear to us how this is defined on general pairs of edges. If α, β areedges corresponding to boundary maps and hence lower the degree by 1, then

161

162 CHAPTER 7. MORE ON DIAGRAMS

we would expect α × β to lower the degree by 2. However, there are no suchedges in H∗Sch′.

Our restricted version of products of diagrams is enough to get the implicationswe want.

In order to control signs in the Kunneth formula, we need to work in a gradedcommutative setting.

Definition 7.1.3. A graded diagram is a diagram D with identities togetherwith a map

| · | : vertices of D → Z/2Z .

For an edge γ : v → v′ we put |γ| = |v| − |v′|. If D is a graded diagram, D ×Dis equipped with the grading |(v, w)| = |v|+ |w|.A commutative product structure on a graded D is a map of graded diagrams

× : D ×D → D

together with choices of edges

αv,w : v × w → w × vβv,w,u : v × (w × u)→ (v × w)× uβ′v,w,u : (v × w)× u→ v × (w × u)

for all vertices v, w, u of D.

A graded multiplicative representation T of a graded diagram with commutativeproduct structure is a representation of T in R−Proj together with a choice ofisomorphism

τ(v,w) : T (v × w)→ T (v)⊗ T (w)

such that:

1. The composition

T (v)⊗ T (w)τ−1(v,w)−−−−→ T (v × w)

T (αv,w)−−−−−→ T (w × v)τ(w,v)−−−−→ T (w)⊗ T (v)

is (−1)|v||w| times the natural map of R-modules.

2. If γ : v → v′ is an edge, then the diagram

T (v × w)T (γ×id)−−−−−→ T (v′ × w)

τ

y yτT (v)⊗ T (w)

(−1)|γ||w|T (γ)⊗id−−−−−−−−−−−−→ T (v′)⊗ T (w)

commutes.

7.1. MULTIPLICATIVE STRUCTURE 163

3. If γ : v → v′ is an edge, then the diagram

T (w × v)T (id×γ)−−−−−→ T (w × v′)

τ

y yτT (w)⊗ T (v)

id⊗T (γ)−−−−−→ T (w)⊗ T (v′)

commutes.

4. The diagram

T (v × (w × u))T (βv,w,u)−−−−−−→ T ((v × w)× u)y y

T (v)⊗ T (w × u) T (v × w)⊗ T (u)y yT (v)⊗ (T (w)⊗ T (u)) −−−−→ (T (v)⊗ T (w))⊗ T (u)

commutes under the standard identification

T (v)⊗ (T (w)⊗ T (u)) ∼= (T (v)⊗ T (w))⊗ T (u).

The maps T (βv,w,u) and T (β′v,w,u) are inverse to each other.

A unit for a graded diagram with commutative product structure D is a vertex1 of degree 0 together with a choice of edges

uv : v → 1× v

for all vertices of v. A graded multiplicative representation is unital if T (1)is free of rank 1 and T (uv) is an isomorphism for all vertices v satisfying thefollowing condition: Let R→ T (1) be the isomorphism determined by

T (u1) : T (1)→ T (1)⊗ T (1).

Under this identification T (uv) identifies with the natural isomorphism

T (v)→ R⊗ T (v).

Remark 7.1.4. 1. In particular, T (αv,w) and T (βv,w,u) are isomorphisms.If v = w then T (αv,v) = (−1)|v|.

2. If 1 is a unit, then T (u1) defines a distinguished isomorphism T (1) →T (1) ⊗ T (1). Hence it is either 0 or a free R-module of rank 1. Thedefinition excluded the first case.


3. Note that the first and the second factor are not treated symmetrically.There is a choice of sign convention involved. The convention above ischosen to be consistent with the one of Section 1.3. Eventually, we want toview relative singular cohomology as graded multiplicative representationin the above sense.

4. For the purposes immediately at hand, the choice of β′v,w,u is not needed.However, it is needed later on in the definition of the product structureon the localized diagram, see Remark 7.2.2.

Let T : D −→ R−Proj be a representation of a diagram with identities. Recallthat we defined its diagram category C(D,T ) (see Definition 6.1.10). It R is afield or a Dedekind domain, then C(D,T ) can be described as the category ofA(D,T )-comodules of finite type over R for the coalgebra A(D,T ) defined inTheorem 6.1.12.

Proposition 7.1.5. Let D be a graded diagram with commutative productstructure with unit and T a unital graded multiplicative representation of Din R−Proj

T : D −→ R−Proj.

1. Then C(D,T ) carries the structure of a commutative and associative ten-sor category with unit and T : C(D,T ) → R−Mod is a tensor functor.On the generators T (v) of C(D,T ) the associativity constraint is inducedby the edges βv,w,u, the commutativity constraint is induced by the edgesαv,w, the unit object is 1 with unital maps induced from the edges uv.

2. If, in addition, R is a field or a Dedekind domain, the coalgebra A(D,T )carries a natural structure of commutative bialgebra (with unit and counit).

The unit object is going to be denoted 1.

Proof. We consider finite diagrams F and F ′ such that

v × w|v, w ∈ F ⊂ F ′ .

We are going to define natural maps

µ∗F : End(T |F ′)→ End(T |F )⊗ End(T |F ).

Assume this for a moment. Let X,Y ∈ C(D,T ). We want to define X ⊗ Y inC(D,T ) = 2−colimFC(F, T ). Let F such that X,Y ∈ C(F, T ). This means thatX and Y are finitely generated R-modules with an action of End(T |F ). Weequip the R-module X ⊗ Y with a structure of End(T |F ′)-module. It is givenby

End(T |F ′)⊗X ⊗ Y → End(T |F )⊗ End(T |F )⊗X ⊗ Y → X ⊗ Y


where we have used the comultiplication map µ∗F and the module structures ofX and Y . This will be independent of the choice of F and F ′. Properties of ⊗on C(D,T ) follow from properties of µ∗F .

If R is a field or a Dedekind domain, let

µF : A(F, T )⊗A(F, T )→ A(F ′, T )

be dual to µ∗F . Passing to the direct limit defines a multiplication µ on A(D,T ).

We now turn to the construction of µ∗F . Let a ∈ End(T |F ′), i.e., a compatiblesystem of endomorphisms av ∈ End(T (v)) for v ∈ F ′. We describe its imageµ∗F (a). Let (v, w) ∈ F × F . The isomorphism

τ : T (v × w)→ T (v)⊗ T (w)

induces an isomorphism

End(T (v × w)) ∼= End(T (v))⊗ End(T (w)).

We define the (v, w)-component of µ∗(a) by the image of av×w under this iso-morphism.

In order to show that this is a well-defined element of End(T |F ) ⊗ End(T |F ),we need to check that diagrams of the form

T (v)⊗ T (w)µ∗(a)(v,w)//

T (α)⊗T (β)

T (v)⊗ T (w)

T (α)⊗T (β)

T (v′)⊗ T (w′)

µ∗(a)(v′,w′)

// T (v′)⊗ T (w′)

commute for all edges α : v → v′, β : w → w′ in F . We factor

T (α)⊗ T (β) = (T (id)⊗ T (β)) (T (α) T (id))

and check the factors separately.

Consider the diagram

T (v × w)av×w

//

T (α×id)

τ

''

T (v × w)

τ

ww

T (α×id)

T (v)⊗ T (w)µ∗(a)(v,w)//

T (α)⊗T (id)

T (v)⊗ T (w)

T (α)⊗T (id)

T (v′)⊗ T (w)

µ∗(a)(v′,w)

// T (v′)⊗ T (w)

T (v′ × w)av′×w //

τ

77

T (v′ × w)

τ

gg


The outer square commutes because a is a diagram endomorphism. Top andbottom commute by definition of µ∗(a). Left and right commute by property(3) up to the same sign (−1)|w||α|. Hence the middle square commutes withoutsigns. The analogous diagram for id× β commutes on the nose. Hence µ∗(a) iswell-defined.

We now want to compare the (v, w)-component to the (w, v)-component. Recallthat there is a distinguished edge αv,w : v × w → w × v. Consider the diagram

T (v)⊗ T (w)µ∗(a)(v,w)//

T (v)⊗ T (w)

T (v × w)

τ

77

T (αv,w)

av×w // T (v × w)

τ

gg

T (αv,w)

T (w × v)

τ''

aw×v // T (w × v)

τww

T (w)⊗ T (v)µ∗(a)(w,v)

// T (w)⊗ T (v)

By the construction of µ∗(a)(v,w) (resp. µ∗(a)(w,v)), the upper (resp. lower)tilted square commutes. By naturality, the middle rectangle with αv,w com-mutes. By property (1) of a representation of a graded diagram with commu-tative product, the left and right faces commute where the vertical maps are(−1)|v||w| times the natural commutativity of tensor products of T -modules.Hence the inner square also commutes without the sign factors. This is cocom-mutativity of µ∗.

The associativity assumption (3) for representations of diagrams with productstructure implies the coassociativity of µ∗.

The compatibility of multiplication and comultiplication is built into the defi-nition.

In order to define a unit object in C(D,T ) it suffices to define a counit forEnd(T |F ). Assume 1 ∈ F . The counit

u∗ : End(T |F ) ⊂∏v∈F

End(T (v))→ End(T (1)) = R

is the natural projection. The assumption on unitality of T allows to check thatthe required diagrams commute.

This finishes the argument for the tensor category and its properties. If R is afield or a Dedekind domain, we have shown that A(D,T ) has a multiplicationand a comultiplication. The unit element 1 ∈ A(D,T ) is induced from thecanonical element 1 ∈ A(1, T ) = EndR(T (1))∨ = R (Note that the lastidentification is indeed canonical, independent of the choice of basis vector in


T (1) ∼= R.) It remains to show that 1 6= 0 in A(D,T ) or equivalently its imageis non-zero in all A(F, T ) with F a finite diagram containing 1. We can view 1as map

End(T |F )→ R .

It is non-zero because it maps id to 1.

Remark 7.1.6. The proof of Proposition 7.1.5 works without any changes inthe arguments when we weaken the assumptions as follows: in Definition 7.1.3replace × by a map of diagrams

× : D ×D → P(D)

where P(D) is the path category of D: objects are the vertices of D and mor-phisms the paths. A representation T of D extends canonically to a functor onP(D).

Example 7.1.7. Let D = N0. We impose a minimal set of edges which allowsfor the definition of a commutative product structure such that n 7→ V ⊗n forfixed vector space V becomes A multiplicative representation. The only edgesare self-edges. We denote them suggestively

ida × αv,w × idb : a+ v + w + b→ a+ w + v + b

with a, b, v, w ∈ N0. We identify ida × α0,0 × idb = ida+b and abbreviate id0 ×αv,w × id0 = αv,w. We turn it into a graded diagram via the trivial grading|n| = 0 fr all n ∈ N.

The summation map

N0 × N0 → N0 (n,m) 7→ n+m

defined a commutative product structure on N0 in the sense of Definition 7.1.3.The definition on edges is the obvious one. All edges βv,w,u, β′v,w,u are given bythe identity. The edges αv,w are the ones specified before. The unit 1 is givenby the vertex 0, the edges uv are given by the identity.

Let V be a finite dimensional k-vector space for some field k. We define a unitalgraded multiplicative representation

T = TV : N0 → k−Mod, n 7→ V ⊗n

The morphisms

τ(v,w) : T (v × w) = V ⊗(n+m)→ T (v)⊗ T (w)

are the natural ones. All conditions are satisfied. We have in particular T (0) =k.

By Proposition 7.1.5, the coalgebra A = A(N0, T ) is a commutative bialgebra.Indeed, SpecA = End(V ) viewed as algebraic monoid over k. In more detail:The commutative algebra A is generated freely by

A(1, T ) = Endk(V )∨.


Let v1, . . . , vn be a basis of v. Then

A(N0, T ) = k[Xij ]ni,j=1

with Xij the element dual to Eij : V → V with Eij(vs) = δisvj . The comul-tiplication A is determined on its value on the Xij where it is induced frommultiplication of the Eij . Hence

∆(Xij) =

n∑s=1

XisXsj .

7.2 Localization

The purpose of this section is to give a diagram version of the localizationof a tensor category with respect to one object, i.e., a distinguished objectX becomes invertible with respect to tensor product. This is the standardconstruction used to pass e.g. from effective motives to all motives.

We restrict to the case when R is a field or a Dedekind domain and all repre-sentations of diagrams take values in R−Proj.

Definition 7.2.1 (Localization of diagrams). Let Deff be a graded diagramwith a commutative product structure with unit 1. Let v0 ∈ Deff be a vertex.The localized diagram D has vertices and edges as follows:

1. for every v a vertex of Deff and n ∈ Z a vertex denoted v(n);

2. for every edge α : v → w in Deff and every n ∈ Z, an edge denotedα(n) : v(n)→ w(n) in D;

3. for every vertex v in Deff and every n ∈ Z an edge denoted (v× v0)(n)→v(n+ 1).

Put |v(n)| = |v|.We equip D with a weak commutative product structure in the sense of Re-mark 7.1.6

× : D ×D → P(D) v(n)× w(m) 7→ (v × w)(n+m)

together with

αv(n),w(m) = αv,w(n+m)

βv(n),w(m),u(r) = βv,w,u(n+m+ r)

β′v(n),w(m),u(r) = β′v,w,u(n+m+ r)

Let 1(0) together withuv(n) = uv(n)

be the unit.

7.2. LOCALIZATION 169

Note that there is a natural inclusion of multiplicative diagrams Deff → D whichmaps a vertex v to v(0).

Remark 7.2.2. The above definition does not spell out × on edges. It isinduced from the product structure on Deff for edges of type (2). For edges oftype (3) there is an obvious sequence of edges. We take their composition inP(D). E.g. for γv,n : (v × v0)(n) → v(n + 1) and idw(m) = idw(m) : w(m) →w(m) we have

γv,n × id(m) : (v × v0)(n)× w(m)→ v(n+ 1)× w(m)

via

(v × v0)(n)× w(m) = ((v × v0)× w)(n+m)

β′v,v0,w(n+m)

−−−−−−−−−→ (v × (v0 × w))(n+m)

id×αv0,w(n+m)−−−−−−−−−−→ (v × (w × v0))(n+m)

βv,w,v0 (n+m)−−−−−−−−−→ ((v × w)× v0)(n+m)

γv×w,n+m−−−−−−−→ (v × w)(n+m+ 1) = v(n+ 1)× w(m) .

Assumption 7.2.3. Let R be a field or a Dedekind domain. Let T be amultiplicative unital representation of Deff with values in R−Proj such thatT (v0) is locally free of rank 1 as R-module.

Lemma 7.2.4. Under Assumption 7.2.3, the representation T extends uniquelyto a graded multiplicative representation of D such that T (v(n)) = T (v) ⊗T (v0)⊗n for all vertices and T (α(n)) = T (α) ⊗ T (id)⊗n for all edges. It ismultiplicative and unital with the choice

T (v(n)× w(m))τv(n),w(m)−−−−−−−→ T (v(n))⊗ T (w(m))

τv,w(n+m)

y y=

T (v)⊗ T (w)⊗ T (v0)⊗n+m∼=−−−−→ T (v)⊗ T (v0)⊗n ⊗ T (w)⊗ T (v0)⊗m

where the last line is the natural isomorphism.

Proof. Define T on the vertices and edges of D via the formula. It is tediousbut straightforward to check the conditions.

Proposition 7.2.5. Let Deff , D and T be as above. Assume Assumption 7.2.3.Let A(D,T ) and A(Deff , T ) be the corresponding bialgebras. Then:

1. C(D,T ) is the localization of the category C(Deff , T ) with respect to theobject T (v0).


2. Let χ ∈ End(T (v0))∨ = A(v0, T ) be the dual of id ∈ End(T (v0)). Weview it in A(Deff , T ). Then A(D,T ) = A(Deff , T )χ (localization of alge-bras).

Proof. Let D≥n ⊂ D be the subdiagram with vertices of the form v(n′) withn′ ≥ n. Clearly, D = colimnD

≥n, and hence

C(D,T ) ∼= 2−colimnC(D≥n, T ) .

Consider the morphism of diagrams

D≥n → D≥n+1, v(m) 7→ v(m+ 1).

It is clearly an isomorphism. We equip C(D≥n+1, T ) with a new fibre functorfT ⊗T (v0)∨. It is faithful exact. The map v(m) 7→ T (v(m+ 1)) is a representa-tion of D≥n in the abelian category C(D≥n+1, T ) with fibre functor fT ⊗T (v0)∨.By the universal property, this induces a functor

C(D≥n, T )→ C(D≥n+1, T ) .

The converse functor is constructed in the same way. Hence

C(D≥n, T ) ∼= C(D≥n+1, T ), A(D≥n, T ) ∼= A(D≥n+1, T ).

The map of graded diagrams with commutative product and unit

Deff → D≥0

induces an equivalence on tensor categories. Indeed, we represent D≥0 inC(Deff , T ) by mapping v(m) to T (v)⊗ T (v0)m. By the universal property (seeCorollary 6.1.18), this implies that there is a faithful exact functor

C(D≥0, T )→ C(Deff , T )

inverse to the obvious inclusion. Hence we also have A(Deff , T ) ∼= A(D≥0, T ) asunital bialgebras.

On the level of coalgebras, this implies

A(D,T ) = colimnA(D≥n, T ) = colimnA(Deff , T )

because A(D≥n, T ) isomorphic to A(Deff , T ) as coalgebras. A(Deff , T ) also has amultiplication, but the A(D≥n, T ) for general n ∈ Z do not. However, they carrya weak A(Deff , T )-module structure analogous to Remark 7.1.6 correspondingto the map of graded diagrams

Deff ×D≥n → P(D≥n).

We want to describe the transition maps of the direct limit. From the point ofview of Deff → Deff , it is given by v 7→ v × v0.

7.3. NORI’S RIGIDITY CRITERION 171

In order to describe the transition maps A(Deff , T ) → A(Deff , T ), it suffices todescribe End(T |F )→ End(T |F ′) where F, F ′ are finite subdiagrams of Deff suchthat v × v0 ∈ V (F ′) for all vertices v ∈ V (F ). It is induced by

End(T (v))→ End(T (v × v0))τ−→ End(T (v))⊗ End(T (v0)) a 7→ a⊗ id.

On the level of coalgebras, this corresponds to the map

A(Deff , T )→ A(Deff , T ), x 7→ xχ.

Note finally, that the direct limit colimA(Deff , T ) with transition maps given bymultiplication by χ agrees with the localization A(Deff , T )χ.

7.3 Nori’s Rigidity Criterion

Implicit in Nori’s construction of motives is a rigidity criterion, which we arenow going to formulate and prove explicitly.

Let R be a Dedekind domain or a field and C an R-linear tensor category. Recallthat R−Mod is the category of finitely generated R-modules and R−Proj thecategory of finitely generated projective R-modules.

We assume that the tensor product on C is associative, commutative and unital.Let 1 be the unit object. Let T : C → R−Mod be a faithful tensor functor withvalues in R−Mod. In particular, T (1) ∼= R. By what we have shown above thisimplies that C is equivalent to the category of representations of a pro-algebraicmonoid over R.

Recall:

Definition 7.3.1. Let C be as a above with R a field. We say that C is rigid,if every object V ∈ C has a strong dual V ∨, i.e., for all X,Y ∈ C

Hom(X ⊗ V, Y ) = Hom(X,V ∨ ⊗ Y ),

Hom(X,V ⊗ Y ) = Hom(X ⊗ V ∨, Y )

By Tannaka duality this implies that the Tannaka dual of C is a group. We aregoing to show below that actually a weaker assumption suffices. Hence by abuseof terminology, we call C rigid also in the case where R is a Dedekind domain,if its Tannaka dual is a group.

We introduce an ad-hoc notion.

Definition 7.3.2. Let V be an object of C. We say that V admits a perfectduality if there is morphism

q : V ⊗ V → 1,

or1→ V ⊗ V


such that T (V ) is projective and T (q) (respectively its dual) is a non-degeneratebilinear form.

Definition 7.3.3. Let V be an object of C. By 〈V 〉⊗ we denote the smallestfull abelian unital tensor subcategory of C containing V .

We start with the simplest case of the criterion.

Lemma 7.3.4. Let V be an object such that C = 〈V 〉⊗ and such that V admitsa perfect duality. Then C is rigid.

Proof. By standard Tannakian formalism, C is the category of comodules for abialgebra A, which is commutative and of finite type as an R-algebra. Indeed:The construction of A as a coalgebra was explained in Proposition 6.1.12. Wemay view C as graded diagram (with trivial grading) with a unital commutativeproduct structure in the sense of Definition 7.1.3. The fibre functor T is a unitalgraded multiplicative representation. The algebra structure on A is the one ofProposition 7.1.5. It is easy to see that A is generated by A(V , T,) as analgebra. The argument is given in more detail below.

We want to show that A is a Hopf algebra, or equivalently, that the algebraicmonoid M = SpecA is an algebraic group.

By Lemma 7.3.7 it suffices to show that there is a closed immersion M → Gof monoids into an algebraic group G. We are going to construct this group orrather its ring of regular functions. We have

A = limAn

with An = A(Cn, T ) for Cn = 〈1, V, V ⊗2, . . . , V ⊗n〉, the smallest full abeliansubcategory containing 1, V, . . . , V ⊗n. By construction, there is a surjectivemap

n⊕i=0

EndR((T (V )⊗i)∨ → An

or, dually, an injective map

A∨n →n⊕i=0

EndR(T (V )⊗i)

where A∨n consists of those endomorphisms compatible with all morphisms inCn. In the limit, there is a surjection of bialgebras

∞⊕i=0

EndR((T (V )⊗i)∨)→ A

and the kernel is generated by the relation defined by compatibility with mor-phisms in C. One such relation is the commutativity constraint, hence the mapfactors via the symmetric algebra

S∗(End(T (V )∨)→ A .

7.3. NORI’S RIGIDITY CRITERION 173

Note that S∗(End(T (V )∨) is canonically the ring of regular functions on thealgebraic monoid End(T (V )). Another morphism in C is the pairing q : V ⊗V →1. We want to work out the explicit equation induced by q.

We choose a basis e1, . . . , er of T (V ). Let

ai,j = T (q)(ei, ej) ∈ R

By assumption, the matrix is invertible. Let Xst be the matrix coefficients onEnd(T (V )) corresponding to the basis ei. Compatibility with q gives for everypair (i, j) the equation

aij = q(ei, ej)

= q((Xrs)ei, (Xr′s′)ej)

= q

(∑r

Xrier,∑r′

Xr′jer′

)=∑r,r′

XriXr′jq(er, er′)

=∑r,r′

XriXr′jarr′

Note that the latter is the (i, j)-term in the product of matrices

(Xir)t(arr′)(Xr′j) .

Let (bij) = (aij)−1. With

(Yij) = (bij)(Xi′r)t(arr′)

we have the coordinates of the inverse matrix. In other words, our set of equa-tions defines the isometry group G(q) ⊂ End(T (V )). We now have expressed Aas quotient of the ring of regular functions of G(q).

The argument works in the same way, if we are given

q : 1→ V ⊗ V

instead.

Proposition 7.3.5 (Nori). Let C and T : C → R−Mod be as defined at thebeginning of the section. Let Vi|i ∈ I be a set of objects of C with the properties:

1. It generates C as an abelian tensor category, i.e., the smallest full abeliantensor subcategory of C containing all Vi is equal to C.

2. For every Vi there is an object Wi and a morphism

qi : Vi ⊗Wi → 1

such that T (qi) : T (Vi) ⊗ T (Wi) → T (1) = R is a perfect pairing of freeR-modules.


Then C is rigid, i.e., for every object V there is a dual object V ∨ such that

Hom(V ⊗A,B) = Hom(A, V ∨⊗B) , Hom(V ∨⊗A,B) = Hom(A, V ⊗B) .

This means that the Tannakian dual of C is not only a monoid but a group.

Remark 7.3.6. The Proposition also holds with the dual assumption, existenceof morphisms

qi : 1→ Vi ⊗Wi

such that T (qi)∨ : T (V )∨ ⊗ T (Wi)

∨ → R is a perfect pairing.

Proof. Consider V ′i = Vi⊕Wi. The pairing qi extends to a symmetric map q′i onV ′i ⊗ V ′i such that T (q′i) is non-degenerate. We now replace Vi by V ′i . Withoutloss of generality, we can assume Vi = Wi.

For any finite subset J ⊂ I, let VJ =⊕

j∈J Vj . Let qJ be the orthogonal sumof the qj for j ∈ J . It is again a symmetric perfect pairing.

For every object V of C, we write 〈V 〉⊗ for the smallest full abelian tensorsubcategory of C containing V . By assumption we have

C =⋃J

〈VJ〉⊗

We apply the standard Tannakian machinery. It attaches to every 〈VJ〉⊗ anR-bialgebra AJ such that 〈VJ〉⊗ is equivalent to the category of AJ -comodules.If we put

A = limAJ

then C will be equivalent to the category of A-comodules. It suffices to showthat AJ is a Hopf-algebra. This is the case by Lemma 7.3.4.

Finally, the missing lemma on monoids.

Lemma 7.3.7. Let R be noetherian ring, G be an algebraic group scheme offinite type over R and M ⊂ G a closed immersion of a submonoid with 1 ∈M(R). Then M is an algebraic group scheme over R.

Proof. This seems to be well-known. It is appears as an exercise in [Re] 3.5.1 2.We give the argument:

Let S be any finitely generated R-algebra. We have to show that the valueS 7→ M(S) is a group. We take base change of the situation to S. Hencewithout loss of generality, it suffices to consider R = S. If g ∈ G(R), we denotethe isomorphism G→ G induced by left multiplication with g also by g : G→ G.Take any g ∈ G(R) such that gM ⊂M (for example g ∈M(R)). Then one has

M ⊇ gM ⊇ g2M ⊇ · · ·

7.4. COMPARING FIBRE FUNCTORS 175

As G is Noetherian, this sequence stabilizes, say at s ∈ N:

gsM = gs+1M

as closed subschemes of G. Since every gs is an isomorphism, we obtain that

M = g−sgsM = g−sgs+1M = gM

as closed subschemes of G. So for every g ∈ M(R) we showed that gM = M .Since 1 ∈M(R), this implies that M(R) is a subgroup.

Example 7.3.8. We explain the simplest example. It is a dressed up versionof Example 7.1.7 where we obtained and algebraic monoid. Let D = N0. Wehave the same self edges ida × αv,w × idb as previously and in addition edgesn+ 2→ n denoted suggestively ida × b× idb : a+ 2 + b→ a+ b.

We equip it with the trivial grading and the commutative product structureobtained by componentwise addition. The unit is given by 0 with uv = id.

Let k be a field and (V, b) a finite dimensional k-vector space with a non-degenerate bilinear form V ×V → k. We define a graded multiplicative rep-resentation

TV,b : N0 → k−Mod v 7→ V ⊗v.

The edge b is mapped to the linear map b : V ⊗2 → k induced from the bilinearmap b. The assumptions of the rigidity criterion in Proposition 7.3.5 are satisfiedfor C = C(D,T ). Indeed it is generated by the object of the form T (1) = V asthe an abelian tensor category. It is self-dual in the sense of the criterion in C.Let v1, . . . , vn be a basis of V and B the matrix of b. The bialgebra A =A(N0, TV,b) is generated by symbols Xij as in Example 7.1.7. There is a relationcoming from the edge b. It was computed in the proof of Lemma 7.3.4 as thematrix product

(Xij)ijB(Xst)st = 0.

Hence

X = SpecA = O(b)

as algebraic group scheme.

7.4 Comparing fibre functors

7.4.1 The space of comparison maps

We pick up the story but with two representations instead of one.

Let R be a Dedekind domain or a field. Let R−Mod be the category of finitelygenerated R-modules and R−Proj the category of finitely generated projec-tive modules. Let D be a graded diagram with commutative product structure


(see Definition 7.1.3) and T1, T2 : D → R−Proj two graded multiplicative rep-resentations. Recall that we have attached coalgebras A1 := A(D,T1) andA2 := A(D,T2) to these representations (see Theorem 6.1.12). They are evenbialgebras by Proposition 7.1.5. The diagram categories C(D,T1) and C(D,T2)are defined as the categories of comodules for these coalgebras. They carry astructure of unital commutative tensor category.

Remark 7.4.1. In the case that D is the diagram defined by a rigid tensorcategory and T1, T2 faithful tensor functors, it is the classical result of Tannakuatheory that not only G1 = SpecA1 and G2 = SpecA2 are both groups, but theyare forms of each other. All tensor functors are isomorphisms and the space ofall tensor functors is a torsor under G1 and G2. Our aim is to imitate this asmuch as possible for a general diagram D. As we will see, the results will beweaker.

Definition 7.4.2. Let D be a diagram, R a Dedekind domain or a field. LetT1 and T2 be representions of D in R−Proj. Let F ⊂ D be a finite subdiagram.We define

Hom(T1|F , T2|F ) =(fp)p∈D ∈∏p∈D

HomR(T1p, T2p)|fq T1m = T2m fp ∀p, q ∈ D ∀m ∈ D(p, q)

.

PutA1,2 = colimFHom(T1|F , T2|F )∨

where ∨ denotes the R-dual and F runs through all finite subdiagrams of D.

Note that our assumptions guarantee that Hom(T1|F , T2|F ) is a projective R-module and hence has a well-behaved R-dual.

Proposition 7.4.3. 1. The operation

End(T1|F )×Hom(T1|F , T2|F )→ Hom(T1|F , T2|F )

induces a compatible comultiplication

A1 ⊗A1,2 ← A1,2.

The operation

Hom(T1|F , T2|F )× End(T2|F )→ Hom(T1|F , T2|F )

induces a compatible comultiplication

A1,2 ⊗A2 ← A1,2.

The composition

Hom(T1|F , T2|F )×Hom(T2|F , T1|F )×Hom(T1|F , T2|F )→ Hom(T1|F , T2|F )→ Hom(T1|F , T2|F )


induces a natural map

A1,2 ⊗A2,1 ⊗A1,2 ← A1,2.

2. Assume that D carries a commutative product structure. Then A1,2 is afaithfully flat commutative unital R-algebra with multiplication induced bythe tensor structure of the diagram category (unless A1,2 = 0) and theabove maps are algebra homomorphisms.

Proof. The statement on comultiplication follows in the same way as the co-multiplication on A1 and A2 themselves, see Theorem 6.1.12. The module A1,2

is faithfully flat over R because it is the direct limit of locally free R-modules.

The hard part is the existence of the multiplication. This follows by goingthrough the proof of Proposition 7.1.5, replacing End(T |F ) by Hom(T1|F , T2|F )in the appropriate places.

Recall that u1 defines a distinguished isomorphism R → Ti(1). The element1 ∈ A1,2 is induced by the image of the map Hom(T1(1), T2(1)) → R dual tothe distinguished basis.

Remark 7.4.4. As in Remark 7.1.6, a weak product structure on D suffices.

Lemma 7.4.5. Let R be a Dedekind domain or a field. Let S be a faithfullyflat ring extension of R. Then the follow data are equivalent:

1. an R-linear map φ∨ : A1,2 → S (of R-algebras);

2. a morphism of representations (with unital commutative product struc-ture). Φ : T1 ⊗ S → T2 ⊗ S;

Moreover, every (unital tensor) functor Φ : C(D,T1)→ C(D,T2) gives rise to amorphism of representations.

Proof. By base change it suffices to consider S = R. This will simplyfiy nota-tion.

We first establish the statement without multiplicative structures. By construc-tion we can restrict to the case where the diagram D is finite.

Such a morphism of representations defines an element φ ∈ Hom(T1, T2) orequvialently an R-linear map φ∨ : A1,2 → R. Conversely, φ a morphism ofrepresentations.

Let Φ : C(D,T1) → C(D,T2) be an S-linear functor. By composing with theuniversal representations T1 and T2 we obtain a morphism of representationsT1⊗ → T2 ⊗ S.

Finally, compatibility with product structure translates into multilicativity ofthe map φ.


Remark 7.4.6. It does not follow that that a morphism of representationsgives rise to functor between categories. Indeed, a linear map V1 → V2 does notgive rise to an algebra homomorphism End(V2)→ End(V1).

We translate the statements to geometric language.

Theorem 7.4.7. Let R be a field or a Dedekind domain. Let D be a diagramwith commutative product structure, T1, T2 : D → R−Proj two representations.Let X1,2 = SpecA1,2, G1 = SpecA1 and G2 = SpecA2. The scheme X1,2 isfaithfully flat over R unless it is empty.

1. The monoid G1 operates on X1,2 from the left

µ : G1 ×X1,2 → X1,2.

2. The monoid G2 operates on X1,2 from the right

µ : X1,2 ×G2 → X1,2.

3. There is a natural morphism

X1,2 ×X2,1 ×X1,2 → X1,2.

Let S be a faithfully flat extension of R. The choice of a point X1,2(S) isequivalent to a morphism of representations T1 ⊗ S → T2 ⊗ S.

Remark 7.4.8. It is possible for X1,2 to be empty as we will see in the examplesbelow.

Example 7.4.9. ForD = Pairs orD = Good and the representations T1 = H∗dR

(de Rham cohomology) and T2 = H∗ (singular cohomology) this is going toinduce the operation of the motivic Galois group Gmot = SpecA2 on the torsorX = SpecA1,2.

We formulate the main result on the comparison of representations. By a torsorwe will mean a torsor in the fpqc-topology, see Definition 1.7.3. For backgroundon torsors, see Section 1.7.

Theorem 7.4.10. Let R→ S be faithfully flat and

ϕ : T1 ⊗R S → T2 ⊗R S

an isomorphism of unitary multiplicative representations.

1. Then there is φ ∈ X1,2(S) such that the induced maps

G1,S → X1,2,S , g 7→ µ(gφ)

G2,S → X1,2,S , g 7→ µ(φg)

are isomorphisms.


2. This map φ induces an equivalence of unital tensor categories

Φ : C(D,T1)→ C(D,T2).

3. The comparison algebra A1,2 is canonically isomorphic to the comparisonalgebra for the category C = C(D,T1) and the fibre functors fT1

and fT2Φ.

Assume in addition that C(D,T1) is rigid. Then:

4. X1,2 is a G1-left torsor and a G2-right torsor in the fpqc-topology.

5. For flat extensions R → S′, all sections ψ ∈ X1,2(S′) are isomorphismsof representations T1 ⊗ S′ → T2 ⊗ S′. The map ψ → ψ−1 defines anisomorphism of schemes ι : X1,2 → X2,1.

6. X1,2 is a torsor in the sense of Definition 1.7.9 with structure map givenby via ι and Theorem 7.4.7

X31,2∼= X1,2 ×X2,1 ×X1,2 → X1,2 .

Moreover, the groups attached to X1,2 via Proposition 1.7.10 are G1 andG2.

Proof. The first statement over S follows directly from the definitions.

We obtain the functor and its inverse by applying the universal property of thediagram catgories in the general form of Corollary 6.1.14. They are inverse toeach other by the uniqueness part of the universal property.

We use the notation A(D,T1, T2) for the period algebra A1,2. By definition,

A(D,T1, T2) = A(D, fT2 Φ T1). The map of diagrams T1 : D → C defines an

algebra homomorphism

A(D,T1, T2)→ A(C, fT1, fT2

Φ)

by the same argument as in the proof of Lemma 6.2.6. We check that it is anisomorphism after base change to S. Over S, we may use the isomorphism φto replace T2 ⊗ S be the isomorphic T1 ⊗ S. The claim now follows from theisomorphism

A(D,T1 ⊗ S)→ A(C(D,T1), fT1)

which is the main content of Theorem 6.1.19 on the diagram category of anabelian category.

Now suppose in addition that C(D,T1) is rigid. By the equivalence this impliesthat C(D,T2) is rigid. This means that the monoids G1 and G2 are groupschemes. The first property translate into X1,2 being a G1-left and G2-righttorsor in the fpqc-topology.

Let ψ : T1 ⊗ S′ → T2 ⊗ S′ be a morphism of representations. We claim thatit is an isomorphism. This can be checked after a base change to S. Then T2


becomes isomorphic to T1 via ϕ and we may replace T2 by T1 in the argument.The morphism ψ can now be identified with a section ψ ∈ G1(S′ ⊗ S). This isa group, hence it has an inverse, which can be interpreted as the inverse of themorphism of representations.

Consider X31,2 → X1,2 as defined in the theorem. We claim that it satisfies the

torsor identities of Definition 1.7.9. This can be checked after base change to Swhere we can replace X1,2 by G1. The map is then given by

G31 → G1, (a, b, c) 7→ ab−1c

which is the trivial torsor. In particular the left group defined by the torsorX1,2 is nothing but G1. The same argument also applies to G2.

Remark 7.4.11. See also the discussion of the Tannakian case in Section 6.1.4.In this case X1,2 is the G-torsor of isomorphisms between the fibre functors T1

and T2 of [DM, Theorem 3.2], see also Theorem 7.4.18. The above theorem ismore general as it starts out with a commutative diagram instead of a rigidcategory. However, it is also weaker as is uses the existence of a point.

7.4.2 Some examples

We make the above theory explicit in a number of simple examples. The aim isto understand conditions needed in order to ensure that X1,2 is a torsor. It willturn out that rigidity of the diagram category is not enough.

Example 7.4.12. We consider again Example 7.1.7. Let k be a field. Thediagram is N0 with only edges ida×αv,w× idb. It carries a commutative productstructure as before.

Let V1 and V2 be finite dimensional k-vector spaces. Let Ti : n 7→ V ⊗ni be themultiplicative representations as in before. We have shown that Gi = End(Vi)as algebraic k-scheme. The same argument yields

X1,2 = Hom(V1, V2)

as algebraic k-scheme with the natural left and right operations by Gi.

Example 7.4.13. We consider again Example 7.3.8. We have D = N0 withadditional edges generated from an extra edge b : 2 → 0. Let (Vi, bi) be finitedimensional vector spaces with a non-degnerate bilinear form. We obtain

X1,2 = Isom((V1, b1), (V2, b2))

the space of linear maps compatible with the forms, i.e., the space of isometries.In this case G1 and G2 are algebraic groups, indeed the orthogonal groups of b1and b2, respectively. The diagram categories were rigid.

We claim that X1,2 = ∅ if dimV2 < dimV1. The argument can already beexplained in the case V1 = k2, V2 = 1 both with the standard scalar product. If


X1,2 6= ∅, there would be a K-valued point for some field extension K/k. Thiswould mean the existence of a linear map K2 → K with matrix (a, b) such thata2 = 1, b2 = 1 and ab = 0. This is impossible. We can write down the sameargument in terms of equations: the algebra A1,2 is generated by X,Y subjectto the equations X2 − 1, Y 2 − 1, XY . This implies 0 = 1 in A1,2.

On the other hand, if dimV1 < dimV2, then X1,2 6= ∅. Nevertheless, the groupsG1, G2 are not isomorphic over any field extension of k. Hence X1,2 is not atorsor. This contrasts starkly to the Tannakian case. Note that the points ofX1,2 do not give rise to functors - they would be tensor functors and henceinvertible.

The example shows:

Corollary 7.4.14. There is a diagram D with unital commuative product struc-ture and a pair of unital multiplicative representations T1, T2 such that the re-sulting tensor categories are both rigid, but non-equivalent.

Example 7.4.15. We resume the situation of Example 7.4.13, but with dimV1 =dimV2. The two spaces become isometric over k because any two non-degeneratebilinear forms are equivalent over the algebraic closure. By Theorem 7.4.10, X1,2

is a torsor and the two diagram categories are equivalent. Hence the categoriesof representations of all orthogonal groups of the same dimension are equivalent.Note that we are considering algebraic k-representations of k-algebraic groupshere.

Example 7.4.16. We consider another variant of Example 7.3.8. Let D = N0

with edges

idn × αv,w × idn : n+ v + w +m→ n+ v + w +m

idn × b× idm : n+ 2 +m→ n+m

idn × b′ × idm : n+m→ n+ 2 +m

with identifications as before idn × α0,0 × idm = idn+m. We use again thetrivial grading and the obvious commutative product structure with all βu,v,wand β′u,v,w given by the identity.

Let (V, b) be a finite dimensional k-vector space with a non-degenerate bilinarform V ⊗2 → k. We define multiplicative representation n 7→ V ⊗n which assignsthe form b to the edge b and the dual of b to the edge b′.

As in the case of Example 7.3.8, the category C(D,T ) is the category of repre-sentations of the group O(b). The algebra is not changed because the additionalrelations for b′ are automatic.

If we have two such representations attached to (V1, b1) and (V2, b2) than X1,2 iseither empty (if dimV1 6= dimV2) or an O(b1)-torsor (if dimV1 = dimV2). Theadditional edge b′ forces any morphism of representations to be an isomorphism.

We formalize this.


Lemma 7.4.17. Let D be graded diagram with a commutative product structure.Let T1, T2 : D → R−Mod be multiplicative representations. Suppose that forevery vertex v there is a vertex w and a pair of edges ev : v × w → 1 ande′v : 1 → v × w such that Ti(ev) and Ti(e

′v) are a non-degenerate bilinear and

its dual.

Let R→ S be faithfully flat. Then every morphism of representations

φ : T1 ⊗ S → T2 ⊗ S

is an isomorphism. Hence Proposition 7.4.10 applies in this case.

As Example 7.4.16 has shown, the space X1,2 may still be empty!

Proof. Let v be an edge. Compatibility with ev forces the map T1(v) ⊗ S →T2(v)⊗S to be injective. Compatibility with e′v forces is to be surjective, hencebijective.

This applies in particular in the Tannakian case. Moreover, in this case X1,2 isnon-empty.

Theorem 7.4.18 (The Tannakian case). Let k be a field, C a rigid tensorcategory. Let F1, F2 : C → k−Mod be two faithful fibre functors with associatedgroups G1 and G2.

1. Let S be a k-algebra. Let

φ : F1 ⊗ S → F2 ⊗ S

be a morphism of tensor functors. Then φ is an isomorphism.

2. X1,2 is non-empty and a G1-left and G2-right torsor.

This is [DM, Proposition 1.9] and [DM, Theorem 3.2]. We give the proof directlyin our notation.

Proof. For the first statement simply apply Proppsition 7.4.17to the diagramdefined by C.We now consider X1,2 and need to show that the natural map k → A1,2 isinjective. As in the proof of Theorem 6.1.19, we can write C = 2−colimp wherep runs through all objects of C. Here where p here means the full subcategorywith only object p. (In general we would consider finite subdiagrams F , but inthe abelian case we can replace F by the direct sum of its objects.) Hence

A1 = limA(p, T1), A1,2 = limA(p, T1, T2).

Without loss of generality we assume that 1 is a direct summand of p.


We check that injectivity on the level of 〈p〉. Let X(p) ⊂ HomR(T1(p), p) bethe object constructed in Lemma 6.3.15. By loc. cit. T1(X(p)) = End(T1|p) =A(p, T1)∨. The same arguments show that

T2(X(p)) = Hom(T1|p, T2|p) = A(〈p〉, T1, T2).

The splitting of p induces a morphism

X(p)→ HomR(T1(p), p)→ HomR(T1(1),1) = 1

Applying T1 gives the map

A(p, T1)∨ → k

defining the unit element of A1. It is surjective. As T1 is faithful, this impliesthat X(p) → 1 is surjective. By applying the faithful functor T2 we get asurjection

A(p, T1, T2)∨ → Homk(T1(1), T2(1)) = k.

This is the map defining the unit of A1,2. Hence k → A1,2 is injective.

7.4.3 The description as formal periods

For later use, we give an alternative description of the same algebra.

Definition 7.4.19. Let D be a diagram. Let T1, T2 : D → R−Proj be represen-tations. We define the space of formal periods P1,2 as the R-module generatedby symbols

(p, ω, γ)

where p is a vertex of D, ω ∈ T1p, γ ∈ T2p∨ with the following relations:

1. linearity in ω, γ;

2. (functoriality) If f : p→ p′ is an edge in D, γ ∈ T2p′∨, ω ∈ T1p, then

(p′, T1f(ω), γ) = (p, ω, T2f∨(γ)).

Proposition 7.4.20. Assume D has a unital commutative product structureand T1, T2 are unital multipliative representation. Then P1,2 is a commutativeR-algebra with multiplication given on generators by

(p, ω, γ)(p′, ω′, γ′) = (p× p′, ω ⊗ ω′, γ ⊗ γ′)

Proof. It is obvious that the relations of P1,2 are respected by the formula.

There is a natural transformation

Ψ : P1,2 → A1,2


defined as follows: let (p, ω, γ) ∈ P1,2. Let F be a finite diagram containing p.Then

Ψ(p, ω, γ) ∈ A1,2(F ) = Hom(T1|F , T2|F )∨,

is the mapHom(T1|F , T2|F )→ Q

which maps φ ∈ Hom(T1|F , T2|F ) to γ(φ(p)(ω)). Clearly this is independent ofF and respects relations of P1,2.

Theorem 7.4.21. Let D be diagram. Then the above map

Ψ : P1,2 → A1,2

is an isomorphism. If D carries a commutative product structure and T1, T2 aregraded multiplicative representations, then it is an isomorphism of R-algebras.

Proof. For a finite subdiagram F ⊂ D let P1,2(F ) be the space of periods. Bydefinition P = colimFP (F ). The statement is compatible with these directlimits. Hence without loss of generality D = F is finite.

By definition, P1,2(F ) is the submodule of∏p∈D

T1p⊗ T2p∨

of elements satisfying the relations induced by edges ofD. By definition, A1,2(F )is the submodule of ∏

p∈DHom(T1p, T2p)

∨

of elements satisfying the relations induced by edges of D. As all Tip are locallyfree and of finite rank, this is the same thing.

The compatibility with products is easy to see.

Remark 7.4.22. The theorem is also of interest in the case T = T1 = T2. Itthen gives an explicit description of Nori’s coalgebra by generators and relations.We have implictly used the description in some of the examples.

Let p be a vertex of D. We choose a basis ω1, . . . , ωn of T1v and a basis γ1, . . . , γnof (T2p)

∨. We callPij = ((p, ωi, γj))i,j

the formal period matrix at p. Will later discuss this point of view systematically.

Proposition 7.4.23. Let D be a diagram with a unital commutative productstructure. Assume that there is a faithfully flat extension R→ S and an isomor-phism of representations ϕ : T1 ⊗ S → T2 ⊗ S. Moreover, assume that C(D,T1)is rigid. Then X1,2 = SpecP1,2 becomes a torsor in the sense of Definition 1.7.9with structure map

P1,2 → P⊗31,2


given by

Pij 7→∑k,`

Pik ⊗ P−1k` ⊗ P`j

Proof. We translate Theorem 7.4.10 into the alternative description.


Chapter 8

Nori motives

We explain Nori’s construction of an abelian category of motives. It is defined asthe diagram category (see Chapters 6 and 7) of a certain diagram. It is universalfor all cohomology theories that can be compared with singular cohomology.

8.1 Essentials of Nori Motives

As before, we denote Z−Mod the category of finitely generated Z-modules andZ−Proj the category of finitely generated free Z-modules.

8.1.1 Definition

Let k be a subfield of C. For a variety X over k, we define singular cohomologyas singular cohomology of X(C) = X ×k C. As in Chapter 2.1, we denote itsimply by Hi(X,Z).

Definition 8.1.1. Let k be a subfield of C. The diagram Pairseff of effectivepairs consists of triples (X,Y, i) with X a k-variety, Y ⊂ X a closed subvarietyand an integer i. There are two types of edges between effective pairs:

1. (functoriality) For every morphism f : X → X ′ with f(Y ) ⊂ Y ′ an edge

f∗ : (X ′, Y ′, i)→ (X,Y, i) .

2. (coboundary) For every chain X ⊃ Y ⊃ Z of closed k-subschemes of Xan edge

∂ : (Y,Z, i)→ (X,Y, i+ 1) .

The diagram has identities (see Definition 6.1.1) given by the identity morphism.The diagram is graded (see Definition 7.1.3) by |(X,Y, i)| = i.

187

188 CHAPTER 8. NORI MOTIVES

Proposition 8.1.2. The assignment

H∗ : Pairseff → Z−Mod

which maps to (X,Y, i) relative singular cohomology Hi(X(C), Y (C),Z) is arepresentation in the sense of Definition 6.1.4. It maps (Gm, 1, 1) to Z.

Proof. Relative singular cohomology was defined in 2.1.1. By definition, it iscontraviantly functorial. This defines H∗ on edges of type 1. The connectingmorphism for triples, see Corollary 2.1.4, defines the representation on edges oftype 2. We compute H1(Gm, 1,Z) via the sequence for relative cohomology

H0(C∗,Z)→ H0(1,Z)→ H1(C∗, 1,Z)→ H1(C∗,Z)→ H1(1,Z)

The first map is an isomorphism. The last group vanishes for dimension reasons.Finally, H1(C∗,Z) ∼= Z because C∗ is homotopy equivalent to the unit circle.

Definition 8.1.3. 1. The category of effective mixed Nori motivesMMeffNori =

MMeffNori(k) is defined as the diagram category C(Pairseff , H∗) from The-

orem 6.1.13.

2. For an effective pair (X,Y, i), we write HiNori(X,Y ) for the corresponding

object in MMeffNori. We put

1(−1) = H1Nori(Gm, 1) ∈MM

effNori ,

the Lefschetz motive.

3. The category MMNori = MMNori(k) of Nori motives is defined as thelocalization of MMeff

Nori with respect to Z(−1).

4. We also write H∗ for the extension of H∗ to MMNori.

Remark 8.1.4. This is equivalent to Nori’s orginal definition by Theorem 8.3.4.

8.1.2 Main results

Theorem 8.1.5 (Nori). 1. MMeffNori has a natural structure of commutative

tensor category with unit such that H∗ is a tensor functor.

2. MMNori is a rigid tensor category.

3. MMNori is equivalent to the category of representations of a pro-algebraicgroup scheme Gmot(k,Z) over Z.

For the proof see Section 8.3.1.

Definition 8.1.6. The group scheme Gmot(k,Z) is called the motivic Galoisgroup in the sense of Nori.

8.1. ESSENTIALS OF NORI MOTIVES 189

Remark 8.1.7. The first statement also holds with the coefficient ring Z re-placed by any noetherian ring R. The other two hold if R is a Dedekind ring Ror field. Of particular interest is the case R = Q.

The proof of this theorem will take the rest of the chapter. We now explain thekey ideas. In order to define the tensor structure, we would like to apply theabstract machine developed in Section 7.1. However, the shape of the Kunnethformula

Hn(X × Y,Q) ∼=⊕i+j=n

Hi(X,Q)⊗Hi(Y,Q)

is not of the required kind. Nori introduces a subdiagram of good pairs whererelative cohomology is concentrated in a single degree and free, so that theKunneth formula simplifies. The key insight now becomes that it is possible torecover all pairs from good pairs. This is done via an algebraic skeletal filtrationconstructed from the Basic Lemma as discussed in Section 2.5. As a byproduct,we will also know that MMeff

Nori and MMNori are given as representations ofan algebra monoid. In the next step, we have to verify rigidity, i.e., we haveto show that the monoid is an algebraic group. We do this by verifying theabstract criterion of Section 7.3.

On the way, we need to establish a general ”motivic” property of Nori motives.

Theorem 8.1.8. There is a natural contravariant triangulated functor

R : Kb(Z[Var])→ Db(MMeffNori)

on the homotopy category of bounded homological complexes in Z[Var] such thatfor every effective pair (X,Y, i) we have

Hi(R(Cone(Y → X)) = HiNori(X,Y ).

For the proof see Section 8.3.1. The theorem allows, for example, to definemotives of simplicial varieties or motives with support.

The category of motives is supposed to be the universal abelian category suchthat all cohomology theories with suitable properties factor via the category ofmotives. We do not yet have such a theory, even though it is reasonable toconjecture that MMNori is the correct description. In any case, it does have auniversal property which is good enough for many applications.

Theorem 8.1.9 (Universal property). Let A be an abelian category with afaithful exact functor f : A → R−Mod for a noetherian ring R. Let

H ′∗ : Pairseff → A

be a representation. Assume that there is an extension R → S such that S isfaithfully flat over R and Z and an isomorphism of representations

Φ : H ′∗S → (f H ′∗)S .


Then H ′∗ extends to MMNori:

Pairseff →MMNori → A[H ′∗(1(−1))]−1.

More precisely, there exists a functor L(H ′∗) :MMNori → A[1(−1)]−1 and anisomorphism of functors

Φ : (fH∗)S → fS L(H ′∗)

such that

MMNori

Pairseff S−Mod

A[H ′∗(1(−1))]−1

H∗

H′∗ fS

(fH∗ )S

H∗S

L(H ′∗)

commutes up to φ and φ. The pair (L(H ′∗), φ) is unique up to unique isomor-phism of functors.

If, moreover, A is a tensor category, f a tensor functor and H ′∗ a gradedmultiplicative representation on Goodeff , then L(H ′∗) is a tensor functor and φis an isomorphism of tensor functors.

For the proof see Section 8.3.1. This means that MMNori is universal for allcohomology theories with a comparison isomorphism to singular cohomology.Actually, it suffice to have a representation of Goodeff or VGoodeff , see Defini-tion 8.2.1.

Example 8.1.10. Let R = k, A = k−Mod, H ′∗ algebraic de Rham cohomologysee Chapter 3. Let S = C, and let the comparison isomorphism Φ be the periodisomorphism of Chapter 5. By the universal property, de Rham cohomologyextends toMMNori. We will study this example in a lot more detail in Part IIIin order to understand the period algebra.

Example 8.1.11. Let R = Z, A the category of mixed Z-Hodge structures,H ′∗ the functor assigning a mixed Hodge structure to a variety or a pair. ThenS = Z and Φ is the functor mapping a Hodge structure to the underlying Z-module. By the universal property, H ′∗ factors canonically via MMNori. Inother words, motives define mixed Hodge structures.

8.2. YOGA OF GOOD PAIRS 191

Example 8.1.12. Let ` be a prime, R = Z`, and A the category of finitelygenerated Z`-modules with a continuous operation of Gal(k/k). Let H ′∗ be `-adic cohomology over k. For X a variety and Y ⊂ X a closed subvariety withopen complement j : U → X, we have

(X,Y, i) 7→ Hiet(Xk, j!Z`).

In this case, we let S = Zl and use the comparison isomorphism between `-adicand singular cohomology.

Corollary 8.1.13. The category MMNori is independent of the choice of em-bedding σ : k → C. More precisely, σ′ : k → C be another embedding. LetH ′∗ be singular cohomology with respect to this embedding. Then there is anequivalence of categories

MMNori(σ)→MMNori(σ′).

Proof. Use S = Z` and the comparison isomorphism given by comparing bothsingular cohomology functors with `-adic cohomology. This induces the functor.

Remark 8.1.14. Note that the equivalence is not canonical. In the argumentabove it depends on the choice of embeddings of k into C extending σ and σ′,respectively. If we are willing to work with rational coefficients instead, we cancompare both singular cohomologies with algebraic de Rham cohomology (withS = k). This gives a compatible system of comparison equivalences.

8.2 Yoga of good pairs

We now turn to alternative descriptions ofMMeffNori better suited to the tensor

structure.

8.2.1 Good pairs and good filtrations

Definition 8.2.1. Let k be a subfield of C.

1. The diagram Goodeff of effective good pairs is the full subdiagram ofPairseff with vertices the triples (X,Y, i) such that singular cohomologysatisfies

Hj(X(C), Y (C);Z) = 0, unless j = i.

and is free for j = i.

2. The diagram VGoodeff of effective very good pairs is the full subdiagramof those effective good pairs (X,Y, i) with X affine, X r Y smooth andeither X of dimension i and Y of dimension i− 1, or X = Y of dimensionless than i.


We will later (see Definition 8.3.2) also introduce the diagrams Pairs of pairs,Good of good pairs and VGood of very good pairs as localization (see Definition7.2.1) with respect to (Gm, 1, 1).

Good pairs exist in abundance by the basic lemma, see Theorem 2.5.2.

Our first aim is to show that the diagram categories attached to Pairseff , Goodeff

and VGoodeff are equivalent. By the general principles of diagram categoriesthis means that we have to represent the diagram Pairseff in C(VGoodeff , H∗).We do this in two steps: a general variety is replaced by the Cech complex at-tached to an affine cover; affine varieties are replaced by complexes of very goodpairs using the key idea of Nori. The construction proceeds in a complicatedway because both steps involve choices which have to be made in a compatibleway. We handle this problem in the same way as in [Hu3].

We start in the affine case. Using induction, one gets from the Basic Lemma 2.5.2:

Proposition 8.2.2. Every affine variety X has a filtration

∅ = F−1X ⊂ F0X ⊂ · · · ⊂ Fn−1X ⊂ FnX = X,

such that (FjX,Fj−1X, j) is very good.

Filtrations of the above type are called very good filtrations.

Proof. Let dimX = n. Put FnX = X. Choose a subvariety of dimension n− 1which contains all singular points of X. By the Basic Lemma 2.5.2, there is asubvariety Fn−1X of dimension n − 1 such that (FnX,Fn−1X,n) is good. Byconstruction Fn−1X r Fn−1X is smooth and hence the pair is very good. Wecontinue by induction.

Corollary 8.2.3. Let X be an affine variety. The inductive system of all verygood filtrations of X is filtered and functorial.

Proof. Let F∗X and F ′∗X be two very good filtrations of X. Fn−1X ∪ F ′n−1Xhas dimension n−1. By the Basic Lemma 2.5.2, there is subvariety Gn−1X ⊂ Xof dimension n− 1 such that (X,Gn−1X,n) is a good pair. It is automaticallyvery good. We continue by induction.

Consider a morphism f : X → X ′. Let F∗X be a very good filtration. Thenf(FiX) has dimension at most i. As in the proof of Corollary 8.2.2, we constructa very good filtration F∗X

′ with the additional property f(FiX) ⊂ FiX ′.

Remark 8.2.4. This allows to construct a functor from the category of affinevarieties to the diagram category C(VGoodeff , H∗) as follows: Given an affinevariety X, let F∗X be a very good filtration. The boundary maps of the triplesFi−1X ⊂ FiX ⊂ Fi+1X define a complex in C(VGoodeff , H∗)

· · · → HiNori(FiX,Fi−1X)→ Hi+1

Nori(Fi+1X,FiX)→ . . .


Taking i-th cohomology of this complex defines an object in C(VGoodeff , H∗)whose underlying Z-module is nothing but singular cohomology Hi(X,Z). Upto isomorphism it is independent of the choice of filtration. In particular, it isfunctorial.

We are going to refine the above construction such that is also applies to com-plexes of varieties.

8.2.2 Cech complexes

The next step is to replace arbitrary varieties by affine ones. The idea for thefollowing construction is from the case of etale coverings, see [F] Definition 4.2.

Definition 8.2.5. Let X a variety. A rigidified affine cover is a finite openaffine covering Uii∈I together with a choice of an index ix for every closedpoint x ∈ X such that x ∈ Uix . We also assume that in the covering every indexi ∈ I occurs as ix for some x ∈ X.

Let f : X → Y be a morphism of varieties, Uii∈I a rigidified open cover of Xand Vjj∈J a rigidified open cover of Y . A morphism of rigidified covers (overf)

φ : Uii∈I → Vjj∈J

is a map of sets φ : I → J such that f(Ui) ⊂ Vφ(i) and for all x ∈ X we haveφ(ix) = jf(x).

Remark 8.2.6. The rigidification makes φ unique if it exists.

Lemma 8.2.7. The projective system of rigidified affine covers is filtered andstrictly functorial, i.e., if f : X → Y is a morphism of varieties, pull-backdefines a map of projective systems.

Proof. Any two covers have their intersection as common refinement with indexset the product of the index sets. The rigidification extends in the obvious way.Preimages of rigidified covers are rigidified open covers.

We need to generalize this to complexes of varieties. Recall from Definition1.1.1 the additive categories Z[Aff] and Z[Var] with objects (affine) varietiesand morphisms roughly Z-linear combinations of morphisms of varieties. Thesupport of a morphism in Z[Var] is the set of morphisms occuring in the linearcombination.

Definition 8.2.8. Let X∗ be a homological complex of varieties, i.e., an objectin Cb(Z[Var]). An affine cover of X∗ is a complex of rigidified affine covers,i.e., for every Xn the choice of a rigidified open cover UXn and for every g :Xn → Xn−1 in the support of the differential Xn → Xn−1 in the complex X∗ amorphism of rigidified covers g : UXn → UXn−1

over g.


Let F∗ : X∗ → Y∗ be a morphism in Cb(Z[Var]) and UX∗ , UY∗ affine covers ofX∗ and Y∗. A morphism of affine covers over F∗ is a morphism of rigidifiedaffine covers fn : UXn → UYn over every morphism in the support of Fn.

Lemma 8.2.9. Let X∗ ∈ Cb(Z[Var]). Then the projective system of rigidifiedaffine covers of X∗ is non-empty, filtered and functorial, i.e., if f∗ : X∗ → Y∗is a morphism of complexes and UX∗ an affine cover of X∗, then there is anaffine cover UY∗ and a morphism of complexes of rigidified affine covers. Anytwo choices are compatible in the projective system of covers.

Proof. Let n be minimal with Xn 6= ∅. Choose a rigidified cover of Xn. Thesupport of Xn+1 → Xn has only finitely many elements. Choose a rigidifiedcover of Xn+1 compatible with all of them. Continue inductively.

Similar constructions show the rest of the assertion.

Definition 8.2.10. Let X be a variety and UX = Uii∈I a rigidified affinecover of X. We put

C?(UX) ∈ C−(Z[Aff]),

the Cech complex associated to the cover, i.e.,

Cn(UX) =∐i∈In

⋂i∈iUi,

where In is the set of tuples (i0, . . . , in). The boundary maps are the onesobtained by taking the alternating sum of the boundary maps of the simplicialscheme.

If X∗ ∈ Cb(Z[Var]) is a complex, and UX∗ a rigidified affine cover, let

C?(UX∗) ∈ C−,b(Z[Aff])

be the double complex Ci(UXj ).

Note that all components of C?(UX∗) are affine. The projective system of thesecomplexes is filtered and functorial.

Definition 8.2.11. Let X be a variety, Uii∈I a rigidified affine cover of X.A very good filtration on UX is the choice of very good filtrations for⋂

i∈JUi

for all J ⊂ I compatible with all inclusions between these.

Let f : X → Y be a morphism of varieties, φ : Uii∈I → Vjj∈J a morphismof rigidified affine covers above f . Fix very good filtrations on both covers. Themorphism φ is called filtered, if for all J ⊂ I the induced map⋂

i∈I′Ui →

⋂i∈I′

Vφ(i)


is compatible with the filtrations.

Let X∗ ∈ Cb(Z[Var]) be a bounded complex of varieties, UX∗ an affine cover ofX∗. A very good filtration on UX∗ is a very good filtration on all UXn compatiblewith all morphisms in the support of the boundary maps.

Note that the Cech complex associated to a rigidified affine cover with verygood filtration is also filtered in the sense that there is a very good filtration onall Cn(UX) and all morphisms in the support of the differential are compatiblewith the filtrations.

Lemma 8.2.12. Let X be a variety, UX a rigidified affine cover. Then theinductive system of very good filtrations on UX is non-empty, filtered and func-torial.

The same statement also holds for a complex of varieties X∗ ∈ Cb(Z[Var]).

Proof. Let UX = Uii∈I be the affine cover. We choose recursively very goodfiltrations on

⋂i∈J Ui with decreasing order of J , compatible with the inclusions.

We extend the construction inductively to complexes, starting with the highestterm of the complex.

Definition 8.2.13. Let X∗ ∈ C−(Z[Aff]). A very good filtration of X∗ is givenby a very good filtration F.Xn for all n which is compatible with all morphismsin the support of the differentials of X∗.

Lemma 8.2.14. Let X∗ ∈ Cb(Z[Var]) and UX∗ an affine cover of X∗ with avery good filtration. Then the total complex of C?(UX∗) carries a very goodfiltration.

Proof. Clear by construction.

8.2.3 Putting things together

Let A be an abelian category with a faithful forgetful functor f : A → R−Modwith R noetherian. Let T : VGoodeff → A be a representation of the diagramof very good pairs.

Definition 8.2.15. Let F•X be an affine variety X together with a very goodfiltration F•. We put R(F•X) ∈ Cb(A)

· · · → T (FjX∗, Fj−1X∗)→ T (Fj+1X∗, FjX∗)→ . . .

Let F•X∗ be a very good filtration of a complex X∗ ∈ C−(Z[Aff]). We putR(F•X∗) ∈ C+(A) the total complex of the double complex R(F.Xn)n∈Z.

Proposition 8.2.16. Let A be an R-linear abelian category with a faithfulforgetful functor f to R−Mod. Let T : VGoodeff → A be a representation such


that f T is singular cohomology with R-coefficients. Then there is a naturalcontravariant triangulated functor

R : Cb(Z[Var])→ Db(A)

on the category of bounded homological complexes in Z[Var] such that for everygood pair (X,Y, i) we have

Hj(R(Cone(Y → X)) =

0 j 6= i,

T (X,Y, i) j = i.

Moreover, the image of R(X) in Db(R−Mod) computes singular cohomology ofX(C).

Proof. We first defineR : Cb(Z[Var])→ Db(A) on objects. LetX∗ ∈ Cb(Z[Var]).Choose a rigidified affine cover UX∗ of X∗. This is possible by Lemma 8.2.9.Choose a very good filtration on the cover. This is possible by 8.2.12. It inducesa very good filtration on TotC?(UX∗). Put

R(X∗) = R(TotC?(UX∗)).

Note that any other choice yields a complex isomorphic to this one in D+(A) be-cause f is faithful and exact and the image of R(X∗) in D+(R−Mod) computessingular cohomology with R-coefficients.

Let f : X∗ → Y∗ be a morphism. Choose a refinement U ′X∗ of UX∗ which

maps to UY∗ and a very good filtration on U ′X∗ . Choose a refinement of the

filtrations on UX∗ and UY∗ compatible with the filtration on U ′X∗ . This gives a

little diagram of morphisms of complexes R which defines R(f) in D+(A).

Remark 8.2.17. Nori suggests working with Ind-objects (or rather pro-objectin our dual setting) in order to get functorial complexes attached to affine va-rieties. However, the mixing between inductive and projective systems in ourconstruction does not make it obvious if this works out for the result we needed.In order to avoid this situation, it might, however, be possible to do the con-struction in two steps. This approach is used in Harrer’s generalization tocomplexes of smooth correspondences, [Ha], which completely avoids discussingCech complexes.

As a corollary of the construction in the proof, we also get:

Corollary 8.2.18. Let X be a variety, UX a rigidified affine cover with Cechcomplex C?(UX). Then

R(X)→ R(C?(UX))

is an isomorphism in D+(A).

We are mostly interested in two explicit examples of complexes.


Definition 8.2.19. Consider the situation of Proposition 8.2.16. Let Y ⊂ Xbe a closed subvariety with open complement U , i ∈ Z. Then we put

R(X,Y ) = R(Cone(Y → X)), RY (X) = R(Cone(U → X)) ∈ Db(A)

H(X,Y, i) = Hi(R(X,Y )), HY (X, i) = Hi(RY (X)) ∈ A

H(X,Y, i) is called relative cohomology. HY (X, i) is called cohomology withsupport.

8.2.4 Comparing diagram categories

We are now ready to proof the first key theorems.

Theorem 8.2.20. The diagram categories C(Pairseff , H∗), C(Goodeff , H∗) andC(VGoodeff , H∗) are equivalent.

Proof. The inclusion of diagrams induces faithful functors

i : C(VGoodeff, H∗)→ C(Goodeff , H∗)→ C(Pairseff , H∗).

We want to apply Corollary 6.1.18. Hence it suffices to represent the diagramPairseff in C(VGoodeff , H∗) such that the restriction of the representation toVGoodeff gives back H∗ (up to natural isomorphism).

We turn to the construction of the representation of Pairseff in C(VGoodeff , H∗).We apply Proposition 8.2.16 to

H∗ : VGoodeff → C(VGoodeff , H∗)

and get a functor

R : Cb(Z[Var])→ Db(C(VGoodeff , H∗)).

Consider an effective pair (X,Y, i) in D. It is represented by

H(X,Y, i) = Hi(R(X,Y )) ∈ C(VGoodeff , H∗)

whereR(X,Y ) = R(Cone(Y → X)) .

The construction is functorial for morphisms of pairs. This allows to representedges of type f∗.

Finally, we need to consider edges corresponding to coboundary maps for triplesX ⊃ Y ⊃ Z. In this case, it follows from the construction of R that there is anatural exact triangle

R(X,Y )→ R(X,Z)→ R(Y, Z).

We use the connecting morphism in cohomology to represent the edge (Y, Z, i)→(X,Y, i+ 1).


For further use, we record a number of corollaries.

Corollary 8.2.21. Every object of MMeffNori is a subquotient of a direct sum of

objects of the form HiNori(X,Y ) for a good pair (X,Y, i) where X = W rW∞

and Y = W0 r (W0 ∩W∞) with W smooth projective, W∞ ∪W0 a divisor withnormal crossings.

Proof. By Proposition 6.1.15, every object in the diagram category of VGoodeff

(and henceMMNori) is a subquotient of a direct sum of some HiNori(X,Y ) with

(X,Y, i) very good. In particular, X r Y can be assumed smooth.

We follow Nori: By resolution of singularities, there is a smooth projectivevariety W and a normal crossing divisor W0∪W∞ ⊂W together with a proper,surjective morphism π : W rW∞ → X such that one has π−1(Y ) = W0 rW∞and π : W r π−1(Y )→ X r Y is an isomorphism. This implies that

H∗Nori(W rW∞,W0 r (W0 ∩W∞))→ H∗Nori(X,Y )

is also an isomorphism by proper base change, i.e., excision.

Remark 8.2.22. Note that the pair (W rW∞,W0 r (W0 ∩W∞) is good, butnot very good in general. Replacing Y by a larger closed subset Z, one may,however, assume that W0 r (W0 ∩W∞) is affine. Therefore, by Lemma 8.3.8,the dual of each generator can be assumed to be very good.

It is not clear to us if it suffices to construct Nori’s category using the diagramof (X,Y, i) with X smooth, Y a divisor with normal crossings. The corollarysays that the diagram category has the right ”generators”, but there might betoo few ”relations”.

Corollary 8.2.23. Let Z ⊂ X be a closed immersion. Then there is a naturalobject Hi

Z(X) in MMNori representing cohomology with supports. There is anatural long exact sequence

· · · → HiZ(X)→ Hi

Nori(X)→ HiNori(X r Z)→ Hi+1

Z (X)→ · · ·

Proof. Let U = X r Z. Put

RZ(X) = R(Cone(U → X)), HiZ(X) = Hi(RZ(X)) .

8.3 Tensor structure

We now introduce the tensor structure using the formal set-up developed inSection 7.1. Recall that Pairseff , Goodeff and VGoodeff are graded diagramswith |(X,Y, i)| = i.

8.3. TENSOR STRUCTURE 199

Proposition 8.3.1. The graded diagrams Good and VGoodeff carry a weakcommutative product structure (see Definition 7.1.3) defined as follows: for allvertices (X,Y, i), (X ′, Y ′, i′)

(X,Y, i)× (X ′, Y ′, i′) = (X ×X ′, X × Y ′ ∪ Y ×X ′, i+ i′).

with the obvious definition on edges. Let also

α : (X,Y, i)× (X ′, Y ′, i′)→ (X ′, Y ′, i′)× (X,Y, i)

β : (X,Y, i)× ((X ′, Y ′, i′)× (X ′′, Y ′′, i′′))→ ((X,Y, i)× (X ′, Y ′, i′))× (X ′′, Y ′′, i′′)

β′ : ((X,Y, i)× (X ′, Y ′, i′))× (X ′′, Y ′′, i′′)→ (X,Y, i)× ((X ′, Y ′, i′)× (X ′′, Y ′′, i′′))

be the edges given by the natural isomorphisms of varieties.

There is a unit given by (Spec k, ∅, 0) and

u : (X,Y, i)→ (Spec k, ∅, 0)× (X,Y, i) = (Spec k ×X,Spec k × Y, i)

be given by the natural isomorphism of varieties.

Moreover, H∗ is a weak graded multiplicative representation (see Definition7.1.3, Remark [?]) with

τ : Hi+i′(X ×X ′, X × Y ′ ∪X ′ × Y,Z)→ Hi(X,Y,Z)⊗Hi′(X ′, Y ′,Z)

the Kunneth isomorphism (see Theorem 2.4.1).

Proof. If (X,Y, i) and (X ′, Y ′, i′) are good pairs, then by the Kuennth formulaso is (X × X ′, X × Y ′ ∪ Y × X ′, i + i′). If they are even very good, then sois their product. Hence × is well-defined on vertices. Recall that edges idGoodeff ×Goodeff are of the form γ× id or id× γ for an edge γ of Goodeff . Thedefinition of × on these edges is the natural one. We explain the case δ × id indetail. Let X ⊃ Y ⊂ Z and A ⊃ B. We compose the functoriality edge for

(Y ×A,Z ×A ∪ Y ×B)→ (Y ×A ∪X ×B,Z ×A ∪ Y ×B)

with the boundary edge for

X ×A ⊃ Y ×A ∪X ×B ⊃ Z ×A ∪ Y ×B

and obtain

δ × id : (Y, Z, n)× (A,B,m) = (Y ×A,Z ×A ∪ Y ×B,n+m)

→ (X ×A, Y ×A ∪X ×B,n+m+ 1) = (X,Y, n+ 1)× (A,B,m)

as a morphism in the path category P(Goodeff).

We need to check that H∗ satisfies the conditions of Definition 7.1.3. This istedious, but straightforward from the properties of the Kunneth formula, see inparticular Proposition 2.4.3 for compatibility with edges of type ∂ changing thedegree.

Associativity and graded commutativity are stated in Proposition 2.4.2.


Definition 8.3.2. Let Good and VGood be the localizations (see Definition7.2.1) of Goodeff and VGoodeff , respectively, with respect to the vertex 1(−1) =(Gm, 1, 1).

Proposition 8.3.3. Good and VGood are graded diagrams with a weak com-mutative product structure (see Remark 7.1.6). Moreover, H∗ is a graded mul-tiplicative representation of Good and VGood.

Proof. This follows formally from the effective case and Lemma 7.2.4. TheAssumption 7.2.3 that H∗(1(−1)) = Z is satisfied by Proposition 8.1.2.

Theorem 8.3.4. 1. This definition ofMMNori is equivalent to Nori’s orig-inal definition.

2. MMeffNori ⊂ MMNori are commutative tensor categories with a faithful

fiber functor H∗.

3. MMNori is equivalent to the digram categories C(Good, H∗) and C(VGood, H∗).

Proof. We already know by Theorem 8.2.20 that

C(VGoodeff , H∗)→ C(Goodeff , H∗)→ C(Pairseff , H∗) =MMeffNori

are equivalent. Moreover, this agrees with Nori’s definition using either Goodeff

or Pairseff .

By Proposition 8.3.1, the diagrams VGoodeff and Goodeff carry a mulitplicativestructure. Hence by Proposition 7.1.5, the category MMeff

Nori carries a tensorstructure.

By Proposition 7.2.5, the diagram categories of the localized diagrams Goodand VGood also have tensor structure and can be equivalently defined as thelocalization with respect to he Lefschetz object 1(−1).

In [L1], the category of Nori motives is defined as the category of comodules offinite type over Z for the localization of the ring Aeff with respect to the elementχ ∈ A(1(−1)) considered in Proposition 7.2.5. By this same Proposition, thecategory of Aeff

χ -comodules agrees with MMNori.

Our next aim is to establish rigidity using the criterion of Section 7.3. Hencewe need to check that Poincare duality is motivic, at least in a weak sense.

Remark 8.3.5. An alternative argument using Harrer’s realization functorfrom geometric motives (see Theorem 10.3.5) is explained in [Ha].

Definition 8.3.6. Let 1(−1) = H1Nori(Gm) and 1(−n) = 1(−1)⊗n.

Lemma 8.3.7. 1. H2nNori(PN ) = 1(−n) for N ≥ n ≥ 0.

2. Let Z be a projective variety of dimension n. Then H2nNori(Z) ∼= 1(−n).


3. Let X be a smooth variety, Z ⊂ X a smooth, irreducible, closed subvarietyof pure codimension n. Then the motive with support of Corollary 8.2.23satisfies

H2nZ (X) ∼= 1(−n).

Proof. 1. Embedding projective spaces linearly into higher dimensional projec-tive spaces induces isomorphisms on cohomology and hence motives. Hence itsuffices to check the top cohomology of PN .

We start with P1. Consider the standard cover of P1 by U1 = A1 and U2 =P1 r 0. We have U1 ∩ U2 = Gm. By Corollary 8.2.18,

R(P1)→ Cone

(R(U1)⊕R(U2)→ R(Gm)

)[−1]

is an isomorphism in the derived category. This induces the isomorphismH2

Nori(P1) → H1Nori(Gm). Similarly, the Cech complex (see Definition 8.2.10)

for the standard affine cover of PN relates H2NNori(PN ) with HN

Nori(GNm).

2. Let Z ⊂ PN be a closed immersion with N large enough. Then H2nNori(Z)→

H2nNori(PN ) is an isomorphism in MMNori because it is in singular cohomology.

3. We note first that under our assumptions 3. holds in singular cohomologyby the Gysin isomorphism 2.1.8

H0(Z)∼=−→H2n

Z (X).

For the embedding Z ⊂ X one has the deformation to the normal cone [Fu, Sec.5.1], i.e., a smooth scheme D(X,Z) together with a morphism to A1 such thatthe fiber over 0 is given by the normal bundle NZX of Z in X, and the otherfibers by X. The product Z × A1 can be embedded into D(X,Z) as a closedsubvariety of codimension n, inducing the embeddings of Z ⊂ X as well as theembedding of the zero section Z ⊂ NZX over 0. Hence, using the three Gysinisomorphisms and homotopy invariance, it follows that there are isomorphisms

H2nZ (X)← H2n

Z×A1(D(X,Z))→ H2nZ (NZX)

in singular cohomology and hence in our category. Thus, we have reduced theproblem to the embedding of the zero section Z → NZX. However, the normalbundle π : NZX → Z trivializes on some dense open subset U ⊂ Z. Thisinduces an isomorphism

H2nZ (NZX)→ H2n

U (π−1(U)),

and we may assume that the normal bundle NZX is trivial. In this case, wehave

NZ(X) = NZ×0(Z × An) = N0(An),

so that we have reached the case of Z = 0 ⊂ An. Using the Kunneth formulawith supports and induction on n, it suffices to consider H2

0(A1) which is

isomorphic to H1(Gm) = 1(−1) by Corollary 8.2.23.


The following lemma (more precisely, its dual) is formulated implicitly in [N] inorder to establish rigidity of MMNori.

Lemma 8.3.8. Let W be a smooth projective variety of dimension i, W0,W∞ ⊂W divisors such that W0 ∪W∞ is a normal crossing divisor. Let

X = W rW∞

Y = W0 rW0 ∩W∞X ′ = W rW0

Y ′ = W∞ rW0 ∩W∞

We assume that (X,Y ) is a very good pair.

Then there is a morphism in MMNori

q : 1→ HiNori(X,Y )⊗Hi

Nori(X′, Y ′)(i)

such that the dual of H∗(q) is a perfect pairing.

Proof. We follow Nori’s construction. The two pairs (X,Y ) and (X ′, Y ′) arePoincare dual to each other in singular cohomology, see Proposition 2.4.5 forthe proof. This implies that they are both good pairs. Hence

HiNori(X,Y )⊗Hi

Nori(X′, Y ′)→ H2i

Nori(X ×X ′, X × Y ′ ∪ Y ×X ′)

is an isomorphism. Let ∆ = ∆(W r (W0 ∪W∞)) via the diagonal map. Notethat

X × Y ′ ∪X ′ × Y ⊂ X ×X ′ r ∆.

Hence, by functoriality and the definition of cohomology with support, there isa map

H2iNori(X ×X ′, X × Y ′ ∪ Y ×X ′)← H2i

∆ (X ×X ′).

Again, by functoriality, there is a map

H2i∆ (X ×X ′)← H2i

∆ (W ×W )

with ∆ = ∆(W ). By Lemma 8.3.7, it is isomorphic to 1(−i). The map qis defined by twisting the composition by (i). The dual of this map realizesPoincare duality, hence it is a perfect pairing.

Theorem 8.3.9 (Nori). MMNori is rigid, hence a neutral Tannakian category.Its Tannaka dual is given by Gmot = Spec(A(Good, H∗)).

Proof. By Corollary 8.2.21, every object of MMNori is a subquotient of M =Hi

Nori(X,Y )(j) for a good pair (X,Y, i) of the particular form occurring inLemma 8.3.8. By this Lemma, they all admit a perfect pairing.

By Proposition 7.3.5, the category MMNori is neutral Tannakian. The Hopfalgebra of its Tannaka dual agrees with Nori’s algebra by Theorem 6.1.20.


8.3.1 Collection of proofs

We go through the list of theorems of Section 8.1 and give the missing proofs.

Proof of Theorem 8.1.5. By Theorem 8.3.4, the categoriesMMeffNori andMMNori

are tensor categories. By construction, H∗ is a tensor functor. The categoryMMNori is rigid by Theorem 8.3.9. By loc. cit., we have a description of itsTannaka dual.

Proof of Theorem 8.1.8. We apply Proposition 8.2.16 with A = MMeffNori and

T = H∗, R = Z.

Proof of Theorem 8.1.9. We apply the universal property of the diagram cate-gory (see Corollary 6.1.14) to the diagram Goodeff , T = H∗ and F = H ′∗. Thisgives the universal property for MMeff

Nori.

Note that H ′∗(1(−1)) ∼= R by comparison with singular cohomology. Henceeverything extends to MMNori by localizing the categories.

If A is a tensor category and H ′∗ a graded multiplicative representation, thenall functors are tensor functors by construction.


Part III

Periods

205

Chapter 9

Periods of varieties

A period, or more precisely, a period number may be thought of as the value ofan integral that occurs in a geometric context. In their papers [K1] and [KZ],Kontsevich and Zagier list various ways of how to define a period.

It is stated in their papers without reference that all these variants give the samedefinition. We give a proof of this statement in the Period Theorem 11.2.1.

9.1 First definition

We start with the simplest definition. In this section, let k ⊂ C be a subfield.

For this definition the following data is needed:

• X a smooth algebraic variety of dimension d, defined over k,

• D a divisor on X with normal crossings, also defined over k,

• ω ∈ Γ(X,ΩdX/k) an algebraic differential form of top degree,

• Γ a rational d-dimensional C∞-chain on Xan with ∂Γ on Dan, i.e.,

Γ =

n∑i=1

αiγi

with αi ∈ Q, γi : ∆d → Xan a C∞-map for all i and ∂Γ a chain on Dan

as in Definition 2.2.2.

As before, we denote by Xan the analytic space attached to X(C).

Definition 9.1.1. Let k ⊂ C be a subfield.

207

208 CHAPTER 9. PERIODS OF VARIETIES

1. Let (X,D, ω,Γ) as above. We will call the complex number∫Γ

ω =

n∑i=1

αi

∫∆d

f∗i ω .

the period (number) of the quadruple (X,D, ω,Γ).

2. The algebra of effective periods Peffnc = Peff

nc (k) over k is the set of all periodnumbers for all (X,D, ω,Γ) defined over k.

3. The period algebra Pnc = Pnc(k) over k is the set of numbers of the form(2πi)nα with n ∈ Z and α ∈ Peff

nc .

Remark 9.1.2. 1. The subscript nc refers to the normal crossing divisor Din the above definition.

2. We will show a bit later (see Proposition 9.1.7) that Peffnc (k) is indeed an

algebra.

3. Moreover, we will see in the next example that 2πi ∈ Peffnc . This means

that Pnc is nothing but the localization

Pnc = Peffnc

[1

2πi

].

4. This definition was motivated by Kontsevich’s discussion of formal effec-tive periods [K1, def. 20, p. 62]. For an extensive discussion of formalperiods and their precise relation to periods see Chapter 12.

Example 9.1.3. Let X = A1Q be the affine line, ω = dt ∈ Ω1. Let D =

V (t3 − 2t). Let γ : [0, 1] → A1Q(C) = C be the line from 0 to

√2. This is a

singular chain with boundary in D(C) = 0,±√

2. Hence it defines a class in

Hsing1 (A1(C)an, Dan,Q). We obtain the period

∫γ

ω =

∫ √2

0

dt =√

2 .

The same method works for all algebraic numbers.

Example 9.1.4. Let X = Gm = A1 \ 0, D = ∅ and ω = 1t dt. We choose

γ : S1 → Gm(C) = C∗ the unit circle. It defines a class in Hsing1 (C∗,Q). We

obtain the period ∫S1

t−1dt = 2πi .

In particular, π ∈ Peffnc (k) for all k.

9.1. FIRST DEFINITION 209

Example 9.1.5. Let X = Gm, D = V ((t − 2)(t − 1)), ω = t−1dt, and γ theline from 1 to 2. We obtain the period∫ 2

1

t−1dt = log(2) .

For more advanced examples, see Part IV.

Lemma 9.1.6. Let (X,D, ω,Γ) as before. The period number∫

Γω depends

only on the cohomology classes of ω in relative de Rham cohomology and of Γin relative singular homology.

Proof. The restriction of ω to the analytification Danj of some irreducible compo-

nent Dj of D is a holomorphic d-form on a complex manifold of dimension d−1,hence zero. Therefore the integral

∫4 ω evaluates to zero for smooth singular

simplices 4 that are supported on D. Now if Γ′, Γ′′ are two representatives ofthe same relative homology class, we have

Γ′d − Γ′′d ∼ ∂(Γd+1)

modulo simplices living on some DanI for a smooth singular chain Γ of dimension

d+ 1Γ ∈ C∞d+1(Xan, Dan;Q).

Using Stokes’ theorem, we get∫Γ′d

ω −∫

Γ′′d

ω =

∫∂(Γd+1)

ω =

∫Γd+1

dω = 0,

since ω is closed.

In the course of the chapter, we are also going to show the converse: every pairof relative cohomology classes gives rise to a period number.

Proposition 9.1.7. The sets Peffnc (k) and Pnc(k) are k-algebras. Moreover,

Peffnc (K) = Peff

nc (k) if K/k is algebraic.

Proof. Let (X,D, ω,Γ) and (X ′, D′, ω′,Γ′) be two quadruples as in the definitionof normal crossing periods.

By multiplying ω by an element of k, we obtain k-multiples of periods.

The product of the two periods is realized by the quadruple (X ×X ′, D×X ′ ∪X ×D′, ω ⊗ ω′,Γ× Γ′).

Note that the quadruple (A1, 0, 1, t., [0, 1]) has period 1. By multiplying withthis factor, we do not change the period number of a quadruple, but we changeits dimension. Hence we can assume that X and X ′ have the same dimension.The sum of their periods is then realized on the disjoint union (X ∪ X ′, D ∪D′, ω + ω′,Γ + Γ′).


If K/k is finite algebraic, then we obviously have Peffnc (k) ⊂ Peff

nc (K). For theconverse, consider a quadruple (X,D, ω,Γ) over K. We may also can view X ask-variety and write Xk for distinction. By Lemma 3.1.13 or more precisely itsproof, ω can also be viewed as a differential form on Xk/k. The complex pointsYk(C) consist of [K : k] copies of the complex points Y (C). Let Γk be the cycleΓ on one of them. Then the period of (X,D, ω,Γ) is the same as the period of(Xk, Dk, ω,Γk). This gives the converse inclusion.

If K/k is infinite, but algebraic, we obviously have Peffnc (K) =

⋃L Peff

nc (L) withL running through all fields K ⊃ L ⊃ k finite over k. Hence, equality also holdsin the general case.

9.2 Periods for the category (k,Q)−Vect

For a clean development of the theory of period numbers, it is of advantageto formalize the data. Recall from Section 5.1 the category (k,Q)−Vect. Itsobjects are a pair of k-vector space Vk and Q-vector space VQ linked by anisomorphism φC : Vk ⊗k C→ VQ ⊗Q C. This is precisely what we need in orderto define periods abstractly.

Definition 9.2.1. 1. Let V = (Vk, VQ, φC) be an object of (k,Q)−Vect. Theperiod matrix of V is the matrix of φC in a choice of bases v1, . . . , vn of Vkand w1, . . . , wn of VQ, respectively. A complex number is a period of V ifit is an entry of a period matrix of V for some choice of bases. The setof periods of V together with the number 0 is denoted P(V ). We denoteby P〈V 〉 the k-subvector space of C generated by the entries of the periodmatrix.

2. Let C ⊂ (k,Q)−Vect be a subcategory. We denote by P(C) the set ofperiods for all objects in C.

Remark 9.2.2. 1. The object V = (Vk, VQ, φC) gives rise to a bilinear map

Vk × V ∨Q → C , (v, λ) 7→ λ(φ−1C (v)) ,

where we have extended λ : VQ → Q C-linearly to VQ ⊗Q C → C. Theperiods of V are the numbers in its image. Note that this image is a set,not a vector space in general. The period matrix depends on the choiceof bases, but the vector space P〈V 〉 does not.

2. The definition of P(C) does not depend on the morphisms. If the categoryhas only one object, the second definition specializes to the first.

Lemma 9.2.3. Let C ⊂ (k,Q)−Vect be a subcategory.

1. P(C) is closed under multiplication by k.

2. If C is additive, then P(C) is a k-vector space.

9.2. PERIODS FOR THE CATEGORY (K,Q)−VECT 211

3. If C is a tensor subcategory, then P(C) is a k-algbra.

Proof. Multiplying a basis element wi by an element α in k multiplies the periodsby α. Hence the set is closed under multiplication by elements of k∗.

Let p be a period of V and p′ a period of V ′. Then p+ p′ is a period of V ⊕ V ′.If C is additive, then V, V ′ ∈ C implies V ⊕ V ′ ∈ C. Moreover, pp′ is a periodof V ⊗ V ′. If C is a tensor subcategory of (k,Q)−Vect, then V ⊗ V ′ is also inC.

Proposition 9.2.4. Let C ⊂ (k,Q)−Vect be a subcategory.

1. Let 〈C〉 be the smallest full abelian subcategory of (k,Q)−Vect closed undersubquotients and containing C. Then P(〈C〉) is the abelian subgroup of Cgenerated by P(C).

2. Let 〈C〉⊗ be the smallest full abelian subcategory of (k,Q)−Vect closedunder subquotients and tensor products and containing C. Then P(〈C〉⊗)is the (possibly non-unital) subring of C generated by P(C).

Proof. The period algebra P(C) only depends on objects. Hence we can replaceC by the full subscategory with the same objects without changing the periodalgebra.

Moreover, if V ∈ C and V ′ ⊂ V in (k,Q)−Vect, then we can extend any basisfor V ′ to a basis to V . In this form, the period matrix for V is block triangularwith one of the blocks the period matrix of V ′. This implies

P(V ′) ⊂ P(V ) .

Hence, P(C) does not change, if we close it up under subobjects in (k,Q)−Vect.The same argument also implies that P(C) does not change if we close it upunder quotients in (k,Q)−Vect.

After these reductions, the only thing missing to make C additive is closing it upunder direct sums in (k,Q)−Vect. If V and V ′ are objects of C, then the periodsof V ⊕ V ′ are sums of periods of V and periods of V ′ (this is most easily seenin the pairing point of view in Remark 9.2.2). Hence closing the category upunder direct sums amounts to passing from P(C) to the abelian group generatedby it. It is automatically a k-vector space.

If V and V ′ are objects of C, then the periods of V ⊗ V ′ are sums of productsof periods of V and periods of V ′ (this is again most easily seen in the pairingpoint of view in Remark 9.2.2). Hence closing C up under tensor products (andtheir subquotients) amounts to passing to the ring generated by P(C).

So far, we fixed the ground field k. We now want to study the behaviour underchange of fields.


Definition 9.2.5. Let K/k be a finite extension of subfields of C. Let

⊗kK : (k,Q)−Vect→ (K,Q)−Vect , (Vk, VQ, φC) 7→ (Vk ⊗k K,VQ, φC)

be the extension of scalars.

Lemma 9.2.6. Let K/k be a finite extension of subfields of C. Let V ∈(k,Q)−Vect. Then

P〈V ⊗k K〉 = P〈V 〉 ⊗k K .

Proof. The period matrix for V agrees with the period matrix for V ⊗k K. Onthe left hand side, we pass to the K-vector space generated by its entries. Onthe right hand side, we first pass to the k-vector space generated by its entries,and then extend scalars.

Conversely, there is a restriction of scalars where we view a K-vector space VKas a k-vector space.

Lemma 9.2.7. Let K/k be a finite extension of subfields of C. Then the functor⊗kK has a right adjoint

RK/k : (K,Q)−Vect→ (k,Q)−Vect

For W ∈ (K,Q)−Vect we have

P〈W 〉 = P〈RK/kW 〉 .

Proof. Choose a k-basis e1, . . . , en of K. We put

RK/k : (K,Q)−Vect→ (k,Q)−Vect , (WK ,WQ, φC) 7→ (WK ,W[K:k]Q , ψC) ,

where

ψC : WK ⊗k C = WK ⊗k K ⊗K C ∼= (WK ⊗K C)[K:k] → (WQ ⊗Q C)[K:k]

maps elements of the form w ⊗ ei to φC(w ⊗ ei) in the i-component.

It is easy to check the universal property. We describe the unit and the counit.The natural map

V → RK/k(V ⊗k K)

is given on the component Vk by the natural inclusion Vk → Vk ⊗K. In orderto describe it on the Q-component, decompose 1 =

∑ni=1 aiei in K and put

VQ → V nQ v 7→ (aiv)ni=1 .

The natural map(RK/kW )⊗k K →W

is given on the K-component as the multiplication map

WK ⊗k K →WK

9.3. PERIODS OF ALGEBRAIC VARIETIES 213

and on the Q-componentWn

Q →WQ

by summation.

This shows existence of the right adjoint. In particular, RK/kW is functorialand independent of the choice of basis.

In order to compute periods, we have to choose bases. Fix a Q-basis x1, . . . , xnof WQ. This also defines a Q-basis for Wn

Q in the obvious way. Fix a K-basisy1, . . . , yn of WK . Multiplying by e1, . . . , en, we obtain a k-basis of WK . Theentries of the period matrix of W are the coefficients of φC(yj) in the basis xlThe entries of the period matrix of RK/kW are the coefficients of φC(eiyj) =in the basis xl. Hence, the K-linear span of the former agrees with the k-linearspan of the latter.

Recall from Example 5.1.4 the object L(α) ∈ (k,Q)−Vect for a complex numberα ∈ C∗. It is given by the data (k,Q, α). It is invertible for the tensor structure.

Definition 9.2.8. Let L(α) ∈ (k,Q)−Vect be invertible. We call a pairing in(k,Q)−Vect

V ×W → L(α)

perfect, if it is non-degenerate in the k- and Q-components. Equivalently, thepairing induces an isomorphism

V ∼= W∨ ⊗ L(α)

where ·∨ denotes the dual in (k,Q)−Vect.

Lemma 9.2.9. Assume that

V ×W → L(α)

is a perfect pairing. Then

P〈V,W, V ∨,W∨〉⊕,⊗ ⊂ P〈V,W 〉⊕,⊗[α−1] .

Proof. The left hand side is the ring generated by P(V ), P(W ), P(V ∨) andP(W∨). Hence we need to show that P(V ∨) and P(W∨) are contained in theright hand side. This is true because W∨ ∼= V ⊗ L(α−1) and P(V ⊗ L(α−1) =α−1P(V )

9.3 Periods of algebraic varieties

9.3.1 Definition

Recall from Definition 8.1.1 the directed graph of effective pairs Pairseff . Itsvertices are triples (X,D, j) with X a variety, D a closed subvariety and j aninteger. The edges are not of importance for the consideration of periods.


Definition 9.3.1. Let (X,D, j) be a vertex of the diagram Pairseff .

1. The set of periods P(X,D, j) is the image of the period paring (see Defi-nition and 5.3.1 and 5.5.4

per : HjdR(X,D)×Hsing

j (Xan, Dan)→ C .

2. In the same situation, the space of periods P〈X,D, j〉 is the Q-vector spacegenerated by P(X,D, j).

3. Let S be a set of vertices in Pairseff(k). We define the set of periods P(S)as the union of the P(X,D, j) for (X,D, j) in S and the k-space of periodsP〈S〉 as the sum of the P〈X,D〉 for (X,D, j) ∈ S.

4. The effective period algebra Peff(k) of k is defined as P(S) for S the set of(isomorphism classes of) all vertices (X,D, j).

5. The period algebra P(k) of k is defined as the set of complex numbers ofthe form (2πi)nα with n ∈ Z and α ∈ Peff(k).

Remark 9.3.2. Note that P(X,D, j) is closed under multiplication by elementsin k but not under addition. However, Peff(k) is indeed an algebra by Corol-lary 9.3.5. This means that P(k) is nothing but the localization

P(k) = Peff(k)

[1

2πi

].

Passing to this localization is very natural from the point of view of motives: itcorresponds to passing from periods of effective motives to periods of all mixedmotives. For more details, see Chapter 10.

Example 9.3.3. Let X = Pnk . Then (Pnk , ∅, 2j) has period set (2πi)jk∗. Theeasiest way to see this is by computing the motive of Pnk , e.g., in Lemma 8.3.7. Itis given by 1(−j). By compatibility with tensor product, it suffices to considerthe case j = 1 where the same motive can be defined from the pair (Gm, ∅, 1).It has the period 2πi by Example 9.1.4. The factor k∗ appears because we maymultiply the basis vector in de Rham cohomology by a factor in k∗.

Recall from Theorem 5.3.3 and Theorem 5.5.6 that we have an explicit descrip-tion of the period isomorphism by integration.

Lemma 9.3.4. There are natural inclusions Peffnc (k) ⊂ Peff(k) and Pnc(k) ⊂

P(k).

Proof. By definition, it suffices to consider the effective case. By Lemma 9.1.6,the period in Peff

nc (k) only depends on the cohomology class. By Theorem 3.3.19,the period in Peff(k) is defined by integration, i.e., by the formula in the defini-tion of Peff

nc (k).

The converse inclusion is deeper, see Theorem 9.4.2.

9.3. PERIODS OF ALGEBRAIC VARIETIES 215

9.3.2 First properties

Recall from Definition 5.4.2 that there is a functor

H : Pairseff → (k,Q)−Vect

where the category (k,Q)−Vect was introduced in Section 5.1. By construction,we have

P(X,D, j) = P(H(X,D, j)),

P〈X,D, j〉 = P〈H(X,D, j)〉,Peff(k) = P(H(Pairseff)) .

This means that we can apply the abstract considerations of Section 5.1 to ourperiods algebras.

Corollary 9.3.5. 1. Peff(k) and P(k) are k-subalgebras of C.

2. If K/k is an algebraic extension of subfields of K, then Peff(K) = Peff(k)and P(K) = P(k).

3. If k is countable, then so is P(k).

Proof. It suffices to consider the effective case. The image of H is closed underdirect sums because direct sums are realized by disjoint unions of effective pairs.As in the proof of Proposition 9.1.7, we can use (A1, 0, 1, 1) in order to shiftthe cohomological degree without changing the periods.

The image of H is also closed under tensor product. Hence its period set is ak-algebra by Lemma 9.2.3.

Let K/k be finite. For (X,D, i) over k, we have the base change (XK , DK , i)over K. By compatibility of the de Rham realization with base change (seeLemma 3.2.14), we have

H(X,D, i)⊗K = H(XK , DK , j) .

By Lemma 9.2.6, this implies that the periods of (X,D, j) are contained in theperiods of the base change. Hence Peff(k) ⊂ Peff(K).

Conversely, if (Y,E,m) is defined over K, we may view it as defined over kvia the map SpecK → Speck. We write (Yk, Ek,m) in order to avoid confu-sion. Note that Yk(C) consists of [K : k] many copies of Y (C). Moreover, byLemma 3.2.15, de Rham cohomology of Y/K agrees with de Rham cohomologyof Yk/k. Hence

H(Yk, Ek,m) = RK/kH(Y,E,m)

and their period sets agree by Lemma 9.2.7. Hence, we also have Peff(K) ⊂Peff(k).


Let k be countable. For each triple (X,D, j), the cohomologies HjdR(X) and

Hsingj (Xan, Dan,Q) are countable. Hence, the image of period pairing is also

countable. There are only countably many isomorphism classes of pairs (X,D, j),hence the set Peff(k) is countable.

9.4 The comparison theorem

We introduce two more variants of period algebras. Recall from Corollary 5.5.2the functor

RΓ : K−(ZSm)→ D+(k,Q)

andHi : K−(ZSm)→ (k,Q)−Vect .

Definition 9.4.1. • Let C(Sm) be the full abelian subcategory of (k,Q)−Vectclosed under subquotients generated by Hi(X•) for X• ∈ K−(ZSm). LetPSm(k) = P(C(Sm)) be the algebra of periods of complexes of smooth va-rieties.

• Let C(SmAff) be the full abelian subcategory of (k,Q)−Vect closed un-der subquotients and generated by Hi(X•) for X• ∈ K−(ZSmAff) withSmAff the category of smooth affine varieties over k. Let PSmAff(k) =P(C(SmAff)) be the algebra of periods of complexes of smooth affine vari-eties.

Theorem 9.4.2. Let k ⊂ C be a subfield. Then all definitions of period algebrasgiven so far agree:

Peff(k) = PSm(k) = PSmAff(k)

andP(k) = Pnc(k) .

Remark 9.4.3. This is a simple corollary of Theorem 8.2.20 and Corollary 8.2.21,once we will have discussed the formal period algebra, see Corollary 12.1.9. How-ever, the argument does not use the full force of Nori’s machine, hence we givethe argument directly. Note that the key input is the same as the key inputinto Nori’s construction: the existence of good filtrations.

Remark 9.4.4. We do not know whether Peff(k) = Peffnc (k). The concrete

definition of Peffnc (k) only admits de Rham classes which are represented by a

global differential form. This is true for all classes in the affine case, but not ingeneral.

Proof. We are going to prove the identities on periods by showing that thesubcategories of (k,Q)−Vect appearing in their definitions are the same.

Let C(Pairseff) be the full abelian subcategory closed under subquotients andgenerated by H(X,D, j) for (X,D) ∈ Pairseff . Furthermore, let C(nc) be the

9.4. THE COMPARISON THEOREM 217

full abelian subcategory closed under subquotients and generated by Hd(X,D)with X smooth, affine of dimension d and D a divisor with normal crossings.

By definitionC(nc) ⊂ C(Pairseff) .

By the construction in Definition 3.3.6, we may compute any H(X,D, j) asHj(C•) with C• in C−(ZSm). Actually, the degree cohomology only dependson a bounded piece of C•. Hence

C(Pairseff) ⊂ C(Sm) .

We next show thatC(Sm) ⊂ C(SmAff) .

Let X• ∈ C−(ZSm). By Lemma 8.2.9, there is a rigidified affine cover UX• ofX•. Let C• = C•(UX•) be the total complex of the associated complex of Chechcomplexes (see Definition 8.2.10). By construction, C• ∈ C−(ZSmAff). By theMayer-Vietoris property, we have

H(X•) = H(C•).

We claim that C(SmAff) ⊂ C(Pairseff). It suffices to consider bounded complexesbecause the cohomology of a bounded above complex of varieties only dependson a bounded quotient. Let X be smooth affine. Recall (see Proposition 8.2.2)that a very good filtration on X is a sequence of subvarieties

F0X ⊂ F1X ⊂ . . . FnX = X

such that FjX r Fj−1X is smooth, with FjX of pure dimension j, or FjX =Fj−1X of dimension less that j and the cohomology of (FjX,Fj−1X) beingconcentrated in degree j. The boundary maps for the triples Fj−2X ⊂ Fj−1X ⊂FjX define a complex R(F.X) in C(Pairseff)

· · · → Hj−1(Fj−1X,Fj−2X)→ Hj(FjX,Fj−1X)→ Hj+1(Fj+1X,FjX)→ . . .

whose cohomology agrees with H•(X).

Let X• ∈ Cb(ZSmAff). By Lemma 8.2.14, we can choose good filtrations on allXn in a compatible way. The double complex R(F.X) has the same cohomologyas X•. By construction, it is a complex in C(Pairseff), hence the cohomology isin C(Pairseff).

Hence, we have now established that

Peffnc (k) ⊂ Peff(k) = PSm(k) = PSmAff(k) .

We refine the argument in order to show that PSmAff(k) ⊂ Pnc(k). By theabove computation, this will follow if periods of very good pairs are containedin Pnc(k). We recall the construction of very good pairs (X,Y, n) by the direct


proof of Nori’s Basic Lemma I in Section 2.5.1. We let X, D0 and D∞ be asin Lemma 2.5.8. In particular, there is a proper surjective map X \D∞ → Xand D0 \ D0 ∩ D∞ = π−1Y . Hence the periods of (X,Y, n) are the same asthe periods of (X \D0, D∞ \D0 ∩D∞, n). The latter cohomology is Poincaredual to the cohomology of the pair (X ′, Y ′, n) = (X \ D∞, D0 \ D0 ∩ D∞, n)by Theorem 2.4.5. In particular, all three are very good pairs with cohomologyconcentrated in degree n and free. Indeed, there is a natural pairing in C

Hd(X,Y )×Hd(X ′, Y ′)→ L((2πi)d).

This is shown by the same arguments as in the proof of Lemma 8.3.8 but withthe functor H instead of Hi

Nori. By Lemma 9.2.9, the periods of (X,Y ) agree upto multiplication by (2πi)d with the periods of (X ′, Y ′). We are now in the situ-ation where X ′ is smooth affine of dimension n and Y ′ is a divisor with normalcrossings. By Proposition 3.3.19, every de Rham cohomology class in degree n isrepresented by a global differential form on X. Hence all cohomological periodsof (X ′, Y ′, n) are normal crossing periods in the sense of Definition 9.1.1.

Chapter 10

Categories of mixed motives

There are different candidates for the category of mixed motives over a field kof characteristic zero. The category of Nori motives of Chapter 8 is one of them.We review two more.

10.1 Geometric motives

We recall the definition of geometrical motives first introduced by Voevodsky,see [VSF] Chapter 5.

As before let k ⊂ C be a field (most of the time suppressed in the notation).

Definition 10.1.1 ([VSF] Chap. 5, Sect. 2.1). The category of finite corre-spondences SmCork has as objects smooth k-varieties and as morphisms fromX to Y the vector space of Q-linear combinations of integral correspondencesΓ ⊂ X × Y which are finite over X and dominant over a component of X.

The composition of Γ : X → Y and Γ′ : Y → Z is defined by push-forward ofthe intersection of Γ × Z and X × Γ′ in X × Y × Z to X × Z. The identitymorphism is given by the diagonal. There is a natural covariant functor

Smk → SmCork

which maps a smooth variety to itself and a morphism to its graph.

The category SmCork is additive, hence we can consider its homotopy categoryKb(SmCork). The latter is triangulated.

Definition 10.1.2 ([VSF] Ch. 5, Defn. 2.1.1). The category of effective geomet-rical motives DM eff

gm = DM effgm(k) is the pseudo-abelian hull of the localization

of Kb(SmCork) with respect to the thick subcategory generated by objects ofthe form

[X × A1 pr→X]

219

220 CHAPTER 10. CATEGORIES OF MIXED MOTIVES

for all smooth varieties X and

[U ∩ V → U q V → X]

for all open covers U ∪ V = X for all smooth varieties X.

Remark 10.1.3. We think of DM effgm as the bounded derived category of the

conjectural abelian category of effective mixed motives.

We denote byM : SmCork → DM eff

gm

the functor which views a variety as a complex concentrated in degree 0. By[VSF] Ch. 5 Section 2.2, it extends (non-trivially!) to a functor on the categoryof all k-varieties.

DM effgm is tensor triangulated such that

M(X)⊗M(Y ) = M(X × Y )

for all smooth varieties X and Y . The unit of the tensor structure is given by

Q(0) = M(Speck) .

The Tate motive Q(1) is defined by the equation

M(P1) = Q(0)⊕Q(1)[2] .

We write M(n) = M ⊗ Q(1)⊗n for n ≥ 0. By [VSF], Chap. 5 Section 2.2, thefunctor

(n) : DM effgm → DM eff

gm

is fully faithful.

Definition 10.1.4. The category of geometrical motives DMgm is the stabi-lization of DM eff

gm with respect to Q(1). Objects are of the form M(n) for n ∈ Zwith

HomDMgm(M(n),M ′(n′)) = HomDMeff

gm(M(n+N),M ′(n′ +N)) N 0 .

Remark 10.1.5. We think of DMgm as the bounded derived category of theconjectural abelian category of mixed motives.

The category DMgm is rigid by [VSF], Chap. 5 Section 2.2, i.e., every objectM has a strong dual M∨ such that

HomDMgm(A⊗B,C) = HomDMgm

(A,B∨ ⊗ C)

A∨ ⊗B∨ = (A⊗B)∨

(A∨)∨ = A

for all objects A,B,C.

10.2. ABSOLUTE HODGE MOTIVES 221

Remark 10.1.6. Rigidity is a deep result. It depends on a moving lemma forcycles and computations in Voevodsky’s category of motivic complexes.

Example 10.1.7. If X is smooth and projective of pure dimension d, then

M(X)∨ = M(X)(−d)[−2d] .

10.2 Absolute Hodge motives

The notion of absolute Hodge motives was introduced by Deligne ([DMOS]Chapter II in the pure case), and independently by Jannsen in ([Ja1]). We followthe presentation of Jannsen, also used in our own extension to the triangulatedsetting ([Hu1]). We give a rough overview over the construction and refer tothe literature for full details.

We fix a subfield k ⊂ C and an algebraic closure k/k. Let Gk = Gal(k/k). LetS be the set of embeddings σ : k → C and S the set of embeddings σ : k → C.Restriction of fields induces a map S → S.

Definition 10.2.1 ([Hu1] Defn. 11.1.1). Let MR = MR(k) be the additivecategory of mixed realizations with objects given by the following data:

• a bifiltered k-vector space AdR;

• for each prime l, a filtered Ql-vector space Al with a continuous operationof Gk;

• for each prime l and each σ ∈ S, a filtered Ql-vector space Aσ,l;

• for each σ ∈ S, a filtered Q-vector space Aσ;

• for each σ ∈ S, a filtered C-vector space Aσ,C;

• for each σ ∈ S, a filtered isomorphism

IdR,σ;AdR ⊗σ C→ Aσ,C ;

• for each σ ∈ S, a filtered isomorphism

Iσ,C : Aσ ⊗Q C→ Aσ,C ;

• for each σ ∈ S and each prime l, a filtered isomorphism

Iσ,l : Aσ ⊗Q Ql → Aσ,l ;

• for each prime l and each σ ∈ S, a filtered isomorphism

Il,σ : Al ⊗Q Ql → Aσ,l .


These data are subject to the following conditions:

• For each σ, the tuple (Aσ, Aσ,C, Iσ,C) is a mixed Hodge structure;

• For each l, the filtration on Al is the filtration by weights: its graded piecesgrWn Al extends to a model of finite type over Z which is pointwise pureof weight n in the sense of Deligne, i.e., for each closed point with residuefield κ, the operation of Frobenius has eigenvalues N(κ)n/2.

Morphisms of mixed realizations are morphisms of this data compatible withall filtrations and comparison isomorphisms.

The above has already used the notion of a Hodge structure as introduced byDeligne.

Definition 10.2.2 (Deligne [D4]). A mixed Hodge structure consists of thefollowing data:

• a finite dimensional filtered Q-vector space (VQ,W∗);

• a finite dimensional bifiltered C-vector space (VC,W∗, F∗);

• a filtered isomorphism IC : (VQ,W∗)⊗ C→ (VC,W∗)

sucht that for all n ∈ Z the induced triple (grWn VQ, grWn VC, grWn I) satisfies

grWn VC =⊕p+q=n

F pgrWn VC ⊕ F qgrnVC

with complex conjugation taken with respect to the R-structure defined bygrWn VQ ⊗ R.

A Hodge structure is called pure of weight n if W∗ is concentrated in degree n.It is called pure if it is direct sum of pure Hodge structures of different weights.

A morphism of Hodge structures are morphisms of this data compatible withfiltration and comparison isomorphism.

By [D4] this is an abelian category. All morphisms of Hodge structures areautomatically strictly compatible with filtrations. This implies immediately:

Proposition 10.2.3 ([Hu1] Lemma 11.1.2). The categoryMR is abelian. Ker-nels and cokernels are computed componentwise.

The notation is suggestive. If X is a smooth variety, then there is a naturalmixed realization H = H∗MR(X) with

• HdR = H∗dR(X) algebraic de Rham cohomology as in Chapter 3 Sec-tion 3.1;

10.2. ABSOLUTE HODGE MOTIVES 223

• Hl = H∗(Xk,Ql) the l-adic cohomology with its natural Galois operation;

• Hσ = H∗(X ×σ Spec(C),Q) singular cohomology;

• Hσ,C = Hσ ⊗ C and Hσ,l = Hσ ⊗Ql;

• IdR,σ is the period isomorphism of Definition 5.3.1 .

• Il,σ is induced by the comparison isomorphism between l-adic and singularcohomology over C.

Remark 10.2.4. If we assume the Hodge or the Tate conjecture, then thefunctor H∗MR is fully faithful on the category of Grothendieck motives (withhomological or, under these assumptions equivalently, numerical equivalence).Hence it gives a linear algebra description of the conjectural abelian category ofpure motives.

Jannsen ([Ja1] Theorem 6.11.1) extends the definition to singular varieties. Arefined version of his construction is given in [Hu1]. We sum up its properties.

Definition 10.2.5 ([Hu2] Defn. 2.2.2). Let C+ be the category with objectsgiven by a tuple of complexes in the additive categories in Definition 10.2.1 withfiltered quasi-isomorphisms between them. The category of mixed realizationcomplexes CMR is the full subcategory of complexes with strict differentialsand cohomology objects inMR. Let DMR be the localization of the homotopycategory of CMR (see [Hu1]) with respect to quasi-isomorphisms (see [Hu1]4.17).

By construction, there are natural cohomology functors:

Hi : CMR →MR

factoring over DMR.

Remark 10.2.6. One should think of DMR as the derived category of MR,even though this is false in a literal sense.

The main construction of [Hu1] is a functor from varieties to mixed realizations.

Theorem 10.2.7 ([Hu1] Section 11.2, [Hu2] Thm 2.3.1). Let Smk be the cate-gory of smooth varieties over k. There is a natural additive functor

RMR : Smk → CMR ,

such thatHiMR(X) = Hi(RMR(X)) .

This allows to extend R to the additive category Q[Smk] and even to the categoryof complexes C−(Q[Smk]).


Remark 10.2.8. There is a subtle technical point here. The category C+ isadditive. Taking the total complex of a complex in C+ gives again an objectof C+. That the subcategory CMR is respected is a non-trivial statement, see[Hu2] Lemma 2.2.5.

Following Deligne and Jannsen, we can now define

Definition 10.2.9. An object M ∈ MR is called an effective absolute Hodgemotive if it is a subquotient of an object in the image of

H∗ R : Cb(Q[Smk])→MR .

Let MMeffAH = MMeff

AH(k) ⊂ MR be the category of all effective absoluteHodge motives over k. Let MMAH = MMAH(k) ⊂ MR be the full abeliantensor subcategory generated by MMeff and the dual of Q(−1) = H2

MR(P1).Objects in MMAH are called absolute Hodge motives over k.

Remark 10.2.10. The rationale behind this definition lies in Remark 10.2.4.Every mixed motive is supposed to be an iterated extension of pure motives.The latter are conjecturally fully described by their mixed realization. Hence,it remains to specify which extensions of pure motives are mixed motives.

Jannsen ([Ja1] Definition 4.1) does not use complexes of varieties but only singlesmooth varietes. It is not clear whether the two definitions agree, see also thediscussion in [Hu1] Section 22.3. On the other hand, in [Hu1] Definition 22.13the varieties were allowed to be singular. This is equivalent to the above by theconstruction in [Hu3] Lemma B.5.3 where every complex of varieties is replacedby complex of smooth varieties with the same cohomology.

Recall the abelian category (k,Q)−Vect from Definition 5.1.1.

Fix ι : k → C. The projection

A 7→ (AdR, Aι, I−1ι,CIdR,ι)

defines a faithful functor

MR→ (k,Q)−Vect .

Recall the triangulated category D+(k,Q) from Definition 5.2.1. The projection

K 7→ (KdR,Kι,Kι,C, IdR,ι, Iι,C)

defines a functorCMR → C+

(k,Q)

which induces also a triangulated functor

forget : DMR → D+(k,Q) .

10.3. COMPARISON FUNCTORS 225

Lemma 10.2.11. There is a natural transformation of functors

K−(Z[Smk])→ D+(k,Q)

between forget RMR and RΓ.

Proof. This is true by construction of the dR- and σ-components of RMR in[Hu1]. In fact, the definition of RΓ is a simplified version of the constructiongiven there. (They are not identical though because MR takes the Hodge andweight filtration into account.)

10.3 Comparison functors

We now have three candidates for categories of mixed motives: the triangulatedcategories of geometric motives and the abelian categories of absolute Hodgemotives and of Nori motives (see Chapter 8).

Theorem 10.3.1. The functor RMR factors via a chain of functors

Cb(Q[Smk])→ DMgm → Db(MMNori)→ Db(MMAH) ⊂ DMR .

The proof will be given at the end of the section. The argument is a bit involved.

Theorem 10.3.2 ([Hu2], [Hu3]). There is a tensor triangulated functor

RMR : DMgm → DMR

such that for smooth X

HiRMR(X) = H∗MR(X) .

For all M ∈ DMgm, the objects HiRMR(M) are absolute Hodge motives.

Proof. This is the main result of [Hu2]. Note that there is a Corrigendum [Hu3].The second assertion is [Hu2] Theorem 2.3.6.

Proposition 10.3.3. Let k ⊂ C.

1. There is a faithful tensor functor

f :MMNori →MMAH

such that the functor RMR : Cb(Q[Smk])→ DMR factors via Db(MMNori)→Db(MMAH).

2. Every object in MMAH is a subquotient of an object in the image ofMMNori.


Proof. We want to use the universal property of Nori motives. Let ι : k ⊂ Cbe the fixed embedding. The assignment A 7→ Aι (see Definition 10.2.1) is afibre functor on the neutral Tannakian category MMAH. We denote it H∗sing

because it agrees with singular cohomology of X ⊗k C on A = H∗MR(X).

We need to verify that the diagram Pairseff of effective pairs from Chap. 8 canbe represented in MMAH in a manner compatible with singular cohomology.More explicitly, let X be a variety and Y ⊂ X a subvariety. Then [Y → X] isan object of DMgm. Hence for every i ≥ 0 there is

HiMR(X,Y ) = HiRMR(X,Y ) ∈MMAH .

By construction, we have

H∗singHiMR(X,Y ) = Hi

sing(X(C), Y (C)) .

The edges in Pairseff are also induced from morphisms in DMgm. Moreover, the

representation is compatible with the multiplicative structure on Goodeff .

By the universal property of Theorem 8.1.9, this yields a functor MMNori →MR. It is faithful, exact and a tensor functor. We claim that it factors viaMMAH. AsMMAH is closed under subquotients inMR, it is enough to checkthis on generators. By Corollary 8.2.21, the category MMeff

Nori is generated byobjects of the form Hi

Nori(X,Y ) for X = W \ W∞ with X smooth and Y adivisor with normal crossings. (In fact, it is generated by very good pairs; blowup the singularities without changing the motive by excision.) Let Y• be theCech nerve of the cover of Y by its normalization. This is the simplicial schemedescribed in detail in Section 3.3.6. Let

C• = Cone(Y• → X)[−1] ∈ C−(Q[Smk]).

Then HiMR(X,Y ) = HiRMR(C•) is an absolute Hodge motive.

Consider X∗ ∈ Cb(Q[Smk]). We apply Proposition 8.2.16 to A = MMNori

and A = MMAH. Hence, there is RNori(X∗) ∈ Db(MMNori) such that theunderlying vector space of HiRNori(X∗) is singular cohomology. We claim thatthere is a natural morphism

f : RNori(X∗)→ RMR(X∗).

It will automatically be a quasi-isomorphism because both compute singularcohomology of X∗.

We continue as in the proof of Proposition 8.2.16. We choose a rigidified affinecover UX∗ of X∗ and a very good filtration on the cover. This induces a verygood filtration on TotC∗(UX∗). This induces a double complex of very goodpairs. Each very good pair may in turn be seen as complex with two entries.We apply RMR to this triple complex and take the associated simple complex.On the one hand, the result is quasi-isomorphic to RMR(X∗) because this istrue in singular cohomology. On the other hand, it agrees with fRNori(X∗), alsoby construction.

10.4. WEIGHTS AND NORI MOTIVES 227

Finally, we claim that every M ∈ MMAH it is subquotient of the image of aNori motive. By definition of absolute Hodge motives it suffices to consider M ofthe form HiRMR(X∗) for X∗ ∈ Cb(Q[Smk]). We have seen that HiRMR(X∗) =Hif(RNori(X∗)), hence M is in the image of f .

Remark 10.3.4. It is very far from clear whether the functor is also full oressentially surjective. The two properties are related because every object inMMAH is a subquotient of an object in the image of MMNori.

Theorem 10.3.5. There is a functor

DMgm → Db(MMNori)

such the composition

Cb(Q[Smk])→ DMgm → Db(MMNori)

agrees with the functor RNori of Proposition 8.2.16.

Proof. This is a result of Harrer, see [Ha].

Proof of Theorem 10.3.1. We put together Theorem 10.3.5 and Theorem 10.3.3.

10.4 Weights and Nori motives

Let k ⊂ C be a subfield. We are now going to explore the connection betweenGrothendieck motives and pure Nori motives and weights.

Definition 10.4.1. Let n ∈ N0. An object M ∈ MMeffNori is called pure of

weight n if it is a subquotient of a motive of the form HnNori(Y ) with Y smooth

and projective.

A motive is called pure if it is a direct sum of pure motives of some weights.

In particular, H∗Nori(Y ) is pure if Y is smooth and projective.

Definition 10.4.2. 1. The category of effective Chow motives CHMeff isgiven by the pseudo-abelian hull of the category with objects given bysmooth, projective varieties and morphism form [X] to [Y ] given by theChow group ChdimX(Y ×X) of algebraic cycles of codimension dimY upto rational equivalence. The category of Chow motives CHM is given bythe localization of the category of effective Chow motives with respect tothe Lefschetz motive L which is the direct complement of [Speck] in P1.

2. The category of effective Grothendieck motives GRMeff is given by thepseudo-abelian hull of the category with objects given by smooth, projec-tive varieties and morphism form [X] to [Y ] given by the group AdimX(Y ×


X) of algebraic cycles of codimension dimY up to homological equiva-lence with respect to singular cohomology. The category of Grothendieckmotives GRM is given by the localization of the category of effectiveGrothendieck motives with respect to the Lefschetz motive L.

In both cases, the composition is given by composition of correspondences.

Remark 10.4.3. There is a contravariant functor X 7→ [X] from the categoryof smooth, projective varieties over k to Chow or Grothendieck motives. It mapsa morphism f : Y → X to the transpose of its graph Γf . The dimension of Γf isthe same as the dimension of Y , hence it has codimension dimX in X ×Y . Onthe other hand, singular cohomology defines a well-defined covariant functor onChow and Grothendieck motives. Note that it is not a tensor functor due tothe signs in the Kunneth formula.

This normalization is the original one, see e.g., [Man]. In recent years, it hasalso become common to use the covariant normalization instead, in particularin the case of Chow motives.

The category of Grothendieck motives is conjectured to be abelian and semi-simple. Jannsen has shown in [Ja2] that this is the case if and only if homologicalequivalence agrees with numerical equivalence.

Proposition 10.4.4. Singular cohomology on GRM factors naturally via afaithful functor

GRM→MMNori

whose image is contained in the category of pure Nori motives.

If the Hodge conjecture holds, then the inclusion is an equivalence of semi-simpleabelian categories.

Proof. The opposite category of CHM is a full subcategory of the category ofgeometric motives DMgm by [VSF, Chapter 5, Proposition 2.1.4]. Restrictingthe contravariant functor

DMgm → Db(MMNori)⊕Hi−−−−→MMNori

to the subcategory yields a covariant functor

CHM→MMNori .

By definition, its image is contained in the category of pure Nori motives. Alsoby definition, a morphism in CHM is zero in GRM if it is zero in singular co-homology, and hence in MMNori. Therefore, the functor automatically factorsvia GRM. The induced functor then is faithful.

We now assume the Hodge conjecture. By [Ja1, Lemma 5.5], this implies thatabsolute Hodge cycles agree with cycles up to homological equivalence. Equiv-alently, the functor GRM → MR to mixed realizations is fully faithful. As itfactors via MMNori, the inclusion GRM→MMNori has to be full as well.

10.5. PERIODS OF MOTIVES 229

The endomorphisms of [Y ] for Y smooth and projective can be computed inMR. Hence it is semi-simple because H∗MR(Y ) is polarizable, see [Hu1, Propo-sition 21.1.2 and 21.2.3]. This implies that its subquotients are the same asits direct summands. Hence, the functor from GRM to pure Nori motives isessentially surjective.

Proposition 10.4.5. Every Nori motive M ∈MMNori carries a unique boundedincreasing filtration (WnM)n∈Z inducing the weight filtration in MR. Everymorphism of Nori motives is strictly compatible with the filtration.

Proof. As the functor MMNori → MR is faithful and exact, the filtration onM ∈ MMNori is indeed uniquely determined by its image in M . Strictness ofmorphisms follows from the same property in MR.

We turn to existence. Bondarko [Bo] constructed what he calls a weight struc-ture on DMgm. It induces a weight filtration on the values of any cohomologicalfunctor. We apply this to the functor to MMNori. In particular, the weightfiltration onHn

Nori(X,Y ) is motivic for every vertex of Pairseff . The weight filtra-tion on subquotients is the induced filtration, hence also motivic. As any objectinMMeff

Nori is a subquotient of some HnNori(X,Y ), this finishes the proof in the

effective case. The non-effective case follows immediately by localization.

10.5 Periods of motives

Recall the chain of functors

DMgm → Db(MMNori)→ Db(MMAH)→ Db((k,Q)−Vect)

constructed in the last section.

Definition 10.5.1. 1. Let C(gm) be the full subcategory of (k,Q)−Vectclosed under subquotients which is generated by H(M) for M ∈ DMgm.Let Pgm = P(C(gm))) be the period algebra of geometric motives.

2. Let C(Nori) be the full subcategory of (k,Q)−Vect closed under subquo-tients which is generated by H(M) for M ∈ MMNori. Let PNori(k) =P(C(Nori)) be the period algebra of Nori motives.

3. Let C(AH) be the full subcategory of (k,Q)−Vect closed under subquo-tients which is generated by H(M) for M ∈ MMAH. Let PAH(k) =P(C(AH)) be the period algebra of absolute Hodge motives.

Proposition 10.5.2. We have

P(k) = Pgm(k) = PNori(k) = PAH(k) .


Proof. From the functors between categories of motives, we have inclusions ofsubcategories of (k,Q)−Vect:

C(gm) ⊂ C(Nori) ⊂ C(AH) .

Moreover, the category C(Smk) of Definition 9.4.1 is contained in C(gm). Bydefinition, we also have C(AH) = C(Smk). Hence, all categories are equal.Finally recall, that P(k) = P(Smk) by Theorem 9.4.2.

This allows easily to translate information on motives into information on peri-ods. Here is an example:

Corollary 10.5.3. Let X be an algebraic space, or, more generally, a Deligne-Mumford stack over k. Then the periods of X are contained in P(k).

Proof. Every Deligne-Mumford stack defines a geometric motive by work ofChoudhury [Ch]. Their periods are therefore contained in the periods of geo-metric motives.

Chapter 11

Kontsevich-Zagier Periods

This chapter follows closely the Diploma thesis of Benjamin Friedrich, see [Fr].The results are due to him.

We work over k = Q or equivalently Q throughout. Denote the integral closureof Q in R by Q. Note that Q is a field.

In this section, we sometimes use X0, ω0 etc. to denote objects over Q and X,ω etc. for objects over C.

11.1 Definition

Recall the notion of a Q-semialgebraic set from Definition 2.6.1.


• G ⊆ Rn be an oriented compact Q-semi-algebraic set which is equidimen-sional of dimension d, and

• ω a rational differential d-form on Rn having coefficients in Q, which doesnot have poles on G.

Then we call the complex number∫Gω a naive period and denote the set of all

naive periods for all G and ω by Pnv.

This set Pnv enjoys additional structure.

Proposition 11.1.2. The set Pnv is a unital Q-algebra.

Proof. Multiplicative structure: In order to show that Pnv is closed under mul-tiplication, we write

pi : Rn1 × Rn2 −→ Rni , i = 1, 2

231

232 CHAPTER 11. KONTSEVICH-ZAGIER PERIODS

for the natural projections and obtain(∫G1

ω1

)·(∫

G2

ω2

)=

∫G1×G2

p∗1ω1 ∧ p∗2ω2 ∈ Pnv

by the Fubini formula.

Multiplication by Q: We find every a ∈ Q as naive period with G = [0, 1] ⊂ Rwith respect to the differential form adt. In particular, 1 ∈ Pnv.

Combining the last two steps, we can shift the dimension of the set G in thedefinition of a period number. Let α =

∫Gω. Represent 1 =

∫[0,1]

dt and

1α =∫G×[0,1]

ω ∧ dt.

Additive structure: Let∫G1ω1 and

∫G2ω2 ∈ Pnv be periods with domains of

integration G1 ⊆ Rn1 and G2 ⊆ Rn2 . Using the dimension shift describedabove, we may assume without loss of generality that dimG1 = dimG2. Usingthe inclusions

i1 : Rn1 ∼= Rn1 × 1/2 × 0 ⊂ Rn1 × R× Rn2 and

i2 : Rn2 ∼= 0 × −1/2 × Rn2 ⊂ Rn1 × R× Rn2 ,

we can write i1(G1) ∪ i2(G2) for the disjoint union of G1 and G2. With theprojections pj : Rn1 ×R×Rn2 → Rnj for j = 1, 2, we can lift ωj on Rnj to p∗jωjon Rn1 × R× Rn2 . For q1, q2 ∈ Q we get

q1

∫G1

ω1+q2

∫G2

ω2 =

∫i1(G1)∪i2(G2)

q1 ·(1/2+t)·p∗1ω1+q2 ·(1/2−t)·p∗2ω2 ∈ Pnv,

where t is the coordinate of the “middle” factor R of Rn1×R×Rn2 . This showsthat Pnv is a Q-vector space.

The Definition 11.1.1 was inspired by the one given in [KZ, p. 772]:

Definition 11.1.3 (Kontsevich-Zagier). A Kontsevich-Zagier period is a com-plex number whose real and imaginary part are values of absolutely convergentintegrals of rational functions with rational coefficients, over domains in Rngiven by polynomial inequalities with rational coefficients.

We will show at the end of this section, that Kontsevich-Zagier periods agreewith naive periods in definition 11.1.1, see Theorem 11.2.4.

Examples of naive periods are

•∫ 2

1

dt

t= log(2),

•∫x2+y2≤ 1

dx dy = π and

11.2. COMPARISON OF DEFINITIONS OF PERIODS 233

•∫G

dt

s=

∫ 2

1

dt√t3 + 1

= elliptic integrals,

for G := (t, s) ∈ R2 | 1 ≤ t ≤ 2, 0 ≤ s, s2 =t3 + 1.

As a problematic example, we consider the following identity.

Proposition 11.1.4 (cf. [K1, p. 62]). We have∫0≤ t1≤ t2≤ 1

dt1 ∧ dt2(1− t1) t2

= ζ(2). (11.1)

Proof. This equality follows by a simple power series manipulation: For 0 ≤t2 < 1, we have ∫ t2

0

dt11− t1

= − log(1− t2) =

∞∑n=1

tn2n.

Let ε > 0. The power series∑∞n=1

tn−12

n converges uniformly for 0 ≤ t2 ≤ 1 − εand we get∫

0≤ t1≤ t2≤ 1−ε

dt1 dt2(1− t1) t2

=

∫ 1−ε

0

∞∑n=1

tn−12

ndt2 =

∞∑n=1

(1− ε)n

n2.

Applying Abel’s Theorem [Fi, XII, 438, 6, p. 411] at (∗), using∑∞n=1

1n3 <∞

gives us ∫0≤ t1≤ t2≤ 1

dt1 dt2(1− t1) t2

= limε→0

∞∑n=1

(1− ε)n

n2

(∗)=

∞∑n=1

1

n2= ζ(2).

Equation (11.1) is not a valid representation of ζ(2) as an integral for a naiveperiod in our sense, because the pole locus t1 = 1 ∪ t2 = 0 of dt1 ∧ dt2

(1−t1) t2is

not disjoint with the domain of integration 0 ≤ t1 ≤ t2 ≤ 1. But (11.1) givesa valid period integral according to the original definition Kontsevich-Zagier —see Definition 11.1.3. We will show in Example 14.1 how to circumvent directlythis difficulty by a blow-up. The general blow-up procedure which makes thispossible is used in the proof of Theorem 11.2.4. This argument shows thatKontsevich-Zagier periods and naive periods are the same.

11.2 Comparison of Definitions of Periods

Theorem 11.2.1 (Friedrich [Fr]).

Peff(Q) = Peffnc (Q) = Peff

nv and P(Q) = Pnc(Q) = Pnv .


The proof will take the rest of this section.

Lemma 11.2.2.Peff

nc (Q) ⊆ Peffnv .

Proof. By definition, its elements of Peffnc (Q) are of the form

∫γω where γ ∈

Hsingd (Xan, Dan,Q) with X a smooth variety of dimension d and D a divisor

with normal crossings and ω ∈ Γ(X,ΩdX).

We choose an embeddingX ⊆ PnQ

(x0:...:xn)

and equip PnQ with coordinates as indicated. Lemma 2.6.5 provides us with amap

ψ : CPn → RN

such that Dan and CPn become Q-semi-algebraic subsets of RN . Then, byProposition 2.6.8, the cohomology class ψ∗γ has a representative which is arational linear combination of singular simplices Γi, each of which is Q-semi-algebraic.

As Peffnv is a Q-algebra by Proposition 11.1.2, it suffices to prove that∫

ψ−1(ImΓi)

ω ∈ Peffnv .

We drop the index i from now. Set G = ImΓ. The claim will be clear as soonas we find a rational differential form ω′ on RN such that ψ∗ω′ = ω, since then∫

ψ−1(G)

ω =

∫ψ−1(G)

ψ∗ω′ =

∫G

ω′ ∈ Peffnv .

After eventually applying a barycentric subdivision to Γ, we may assume w.l.o.g.that there exists a hyperplane in CPn, say x0 = 0, which does not meetψ−1(G). Furthermore, we may assume that ψ−1(G) lies entirely in Uan for Uan open affine subset of D ∩ x0 6= 0. (As usual, Uan denotes the complexanalytic space associated to the base change to C of U .) The restriction of ω tothe open affine subset can be represented in the form (cf. [Ha2, II.8.4A, II.8.2.1,II.8.2A]) ∑

|J|=d

fJ(x0, . . . , xn) d

(xj1x0

)∧ . . . ∧ d

(xjdx0

)with fJ(x1, · · · , xn) ∈ Q(x0, · · · , xn) being homogenous of degree zero. Thisexpression defines a rational differential form on all of PnQ with coefficients in Qand it does not have poles on ψ−1(G).

We construct the rational differential form ω′ on RN with coefficients in Q(i)as follows

ω′I :=∑|J|=d

fJ

(1,y10 + iz10

y00 + iz00, · · · , yn0 + izn0

y00 + iz00

)d

(yj10 + izj10

y00 + iz00

)∧ . . .∧ d

(yjd0 + izjd0

y00 + iz00

),


where we have used the notation from the proof of Lemma 2.6.5. Using theexplicit form of ψ given in this proof, we obtain

ψ∗fJ

(1,y10 + iz10

y00 + iz00, · · · , yn0 + izn0

y00 + iz00

)= fJ

(x0x0

|x0|2,x1x0

|x0|2, . . . ,

xnx0

|x0|2

)= fJ(x0, x1, . . . , xn)

and

ψ∗d

(yj0 + izj0y00 + iz00

)= d

(xjx0

|x0|2

)= d

(xjx0

).

This shows that ψ∗ω′ = ω and we are done.

Lemma 11.2.3.Peff

nv ⊆ Peffnc (Q) .

Proof. We will use objects over various base fields. We will use subscripts toindicate which base field is used: A 0 for Q, a 1 for Q, a subscript R for R andnone for C. Furthermore, we fix an embedding Q ⊂ C.

Let∫GωR ∈ Pnv be a naıve period with

• G ⊂ Rn an oriented Q-semi-algebraic set, equidimensional of dimensiond, and

• ωR a rational differential d-form on Rn with coefficients in Q, which doesnot have poles on G.

The Q-semi-algebraic set G ⊂ Rn is given by polynomial inequalities and equal-ities. By omitting the inequalities but keeping the equalities in the definitionof G, we see that G is supported on (the set of R-valued points of) a variety

YR ⊆ AnR of same dimension d. This variety YR is already defined over Q

YR = Y0 ×Q R

for a variety Y0 ⊆ AnQ

over Q. Similarly, the boundary ∂G of G is supported on

a variety ER, likewise defined over Q

ER = E0 ×Q R.

Note that E0 is a divisor on Y0. By eventually enlarging E0, we may assumew.l.o.g. that E0 contains the singular locus of Y0. In order to obtain an abstractperiod, we need smooth varieties. The resolution of singularities according toHironaka [Hi1] provides us with a Cartesian square

E0 ⊆ Y0

↓ ↓ π0

E0 ⊆ Y0

(11.2)

where


• Y0 is smooth and quasi-projective,

• π0 is proper, surjective and birational, and

• E0 is a divisor with normal crossings.

In fact, π0 is an isomorphism away from E0 since the singular locus of Y0 iscontained in E0

π0|U0: U0

∼−→ U0 (11.3)

with U0 := Y0 \ E0 and U0 := Y0 \ E0.

We apply the analytification functor to the base change to C of the map π0 :Y0 → Y0 and obtain a projection

πan : Y an → Y an.

We want to show that the “strict transform” of G

G := π−1an (G \ Ean) ⊆ Y an

can be triangulated. Since CPn is the projective closure of Cn, we have Cn ⊂CPn and thus get an embedding

Y an ⊆ Cn ⊂ CPn.

We also choose an embedding

Y an ⊆ CPm

for some m ∈ N. Using Lemma 2.6.5, we may consider both Y an and Y an asQ-semi-algebraic sets via some maps

ψ : Y an ⊂ CPn → RN , and

ψ : Y an ⊆ CPm → RM .

In this setting, the induced projection

πan : Y an −→ Y an

becomes a Q-semi-algebraic map. The composition of ψ with the inclusionG ⊆ Y an is a Q-semi-algebraic map; hence G ⊂ RN is Q-semi-algebraic by Fact2.6.4. Since Ean is also Q-semi-algebraic via ψ, we find that G \Ean is Q-semi-

algebraic. Again by Fact 2.6.4, π−1an (G \ Ean) ⊂ RM is Q-semi-algebraic. Thus

G ⊂ RM , being the closure of a Q-semi-algebraic set, is Q-semi-algebraic. FromProposition 2.6.8, we see that G can be triangulated

G = ∪j4j , (11.4)

where the 4j are (homeomorphic images of) d-dimensional simplices.


Our next aim is to define an algebraic differential form ω1 replacing ωR. Wefirst make a base change in (11.2) from Q to Q and obtain

E1 ⊆ Y1

↓ ↓ π1

E1 ⊆ Y1 .

The differential d-form ωR can be written as

ωR =∑|J|=d

fJ(x1, . . . , xn) dxj1 ∧ . . . ∧ dxjd , (11.5)

where x1, . . . , xn are coordinates of Rn and fJ ∈ Q(x1, . . . , xn). We can useequation (11.5) to define a differential form ω1 on AnQ

ω1 =∑|J|=d

fJ(x1, . . . , xn) dxj1 ∧ . . . ∧ dxjd ,

where now x1, . . . , xn denote coordinates of AnQ. The pole locus of ω1 gives us

a variety Z1 ⊂ AnQ. We set

X1 := Y1 \ Z1, D1 := E1 \ Z1, and

X1 := π−11 (X1), D1 := π−1

1 (D1).

The restriction ω1|X1of ω1 to X1 is a (regular) algebraic differential form on

X1; the pullbackω1 := π∗1(ω1|X1

)

is an algebraic differential form on X1.

We consider the complex analytic spaces Xan, Dan, Zan associated to the basechange to C of X1, D1, Z1. Since ω1 has no poles on G, we have G ∩ Zan = ∅;hence G ∩ π−1

an (Zan) = ∅. This shows G ⊆ X = Y \ π−1an (Zan).

Since G is oriented, so is π−1an (G \ Ean), because πan is an isomorphism away

from Ean. Every d-simplex 4j in (11.4) intersects π−1an (G \ Ean) in a dense

open subset, hence inherits an orientation. As in the proof of Proposition 2.6.8,we choose orientation-preserving homeomorphisms from the standard d-simplex4stdd to 4j

σj : 4stdd −→ 4j .

These maps sum up to a singular chain

Γ = ⊕j σj ∈ Csingd (Xan;Q).

It might happen that the boundary of the singular chain Γ is not supported on∂G. Nevertheless, it will always be supported on Dan: The set π−1

an (G \Ean) isoriented and therefore the boundary components of ∂4j that do not belong to


∂G cancel if they have non-zero intersection with π−1an (G \ Ean). Thus Γ gives

rise to a singular homology class

γ ∈ Hsingd (Xan, Dan;Q).

We denote the base change to C of ω1 and ω1 by ω and ω, respectively. Now∫G

ω1 =

∫G

ω =

∫G∩Uan

ω

(11.3)=

∫π−1(G∩Uan)

π∗ω =

∫G∩Uan

ω

=

∫G

ω =

∫Γ

ω =

∫γ

ω ∈ Peffnc (Q)

is a period for the quadruple (X1, D1, ω1, γ).

Proof of Theorem 11.2.1. It suffices to consider the effective case. By Theorem9.4.2, we have Peff(Q) = Peff

nc (Q). By Corollary 9.3.5, this is also the sameas Peff(Q). The result now follows by combining Lemma 11.2.2 and Lemma11.2.3.

Now, we show that naive periods and Kontsevich-Zagier periods coincide:

Theorem 11.2.4.

PeffKZ = Peff

nv = Peff , PKZ = Pnv = P.

Proof. We will use that Peffnv = Peff

nc = Peff (see Theorem 11.2.1) and work witheffective periods only. We partially follow ideas of Belkale and Brosnan [BB].First we show that Peff

KZ ⊆ Peffnc : Assume we have given a period through an

n-dimensional absolutely convergent integral∫

∆ω, where ω = f(x1,...,xn)

g(x1,...,xn) is a

rational function defined over Q and ∆ a Q-semialgebraic region defined by in-equalities hi ≥ 0. This defines a rational differential form ω on An. We canextend ω to a rational differential form on Pn (also denoted by ω) by addinga homogenous variable x0. The closure ∆ of ∆ in Pn(R) is a compact semial-gebraic region, defined by Hi ≥ 0 for some homogenous polynomials Hi. LetH =

∏iHi. Now we use resolution of singularities and obtain a blow-up

σ : X → Pn,

such that we have the following properties:1. σ is an isomorphism outside the union of the pole locus of ω and the zerosets of all polynomials Hi.2. The strict transform of the zero locus of H is a normal crossing divisor in X.


3. Near each point P ∈ X, there are local algebraic coordinates x1, ..., xn andintegers ej , fj for each j = 1, ..., n, such that

H σ = unit1 ×n∏j=1

xejj , σ

∗ω = unit2 ×n∏j=1

xfjj dx1 ∧ · · · ∧ dxn.

Let ∆ be the analytic closure of ∆ ∩ U , where U is the set where σ is anisomorphism. Then ∆ is compact, since it is a closed subset of the compact setσ−1(∆). The absolute convergence of

∫∆ω implies the local convergence of σ∗ω

over regions 0 < xi < ε at point P ∈ ∆. This is only possible, if all fj ≥ 0.Therefore, σ∗ω is regular (holomorphic) at the point P , and hence on the wholeof ∆.

Now we show that Peffnc ⊆ Peff

KZ: This argument is indicated in Kontsevich-Zagier[KZ, pg. 773]. First, note that naive periods in Peff

KZ can also be defined with Q-coefficients and the polynomials involved can be replaced by algebraic functionswithout changing the set Peff

KZ. A proof is not given in loc. cit., but this can beachieved by using auxiliary variables and minimal polynomials as in the proofthat

√2 ∈ Peff

KZ. Assuming this, we now assume that we have given a smoothalgebraic variety X of dimension n, a regular differential from ω of top degree(hence closed), a normal crossing divisor D ⊂ X, all this data defined over Q,and a singular chain γ with boundary ∂γ ⊂ D. Now we can use the method ofLemma 11.2.2 and we can write ∫

γ

ω =

∫G

ω,

where G is a Q-semialgebraic subset of the required form, i.e., given by inequal-ities, and ω is a differential form with algebraic coefficients.


Chapter 12

Formal periods and theperiod conjecture

Following Kontsevich (see [K1]), we now introduce another algebra P(k) of for-mal periods from the same data we have used in order to define the actual periodalgebra of a field in Chapter 9. It comes with an obvious surjective map to P(k).

The first aim of the chapter is to give a conceptual interpretation of P(k) asthe ring of algebraic functions on the torsor between two fibre functors on Norimotives: singular cohomomology and algebraic de Rham cohomology.

We then discuss the period conjecture from this point of view.

12.1 Formal periods and Nori motives

Definition 12.1.1. Let k ⊂ C be a subfield. The space of effective formal pe-riods Peff(k) is defined as the Q-vector space generated by symbols (X,D, ω, γ),where X is an algebraic variety over k, D ⊂ X a subvariety, ω ∈ Hd

dR(X,D),γ ∈ Hd(X(C), D(C),Q) with relations

1. linearity in ω and γ;

2. for every f : X → X ′ with f(D) ⊂ D′

(X,D, f∗ω′, γ) = (X ′, D′, ω′, f∗γ)

3. for every triple Z ⊂ Y ⊂ X

(Y,Z, ω, ∂γ) = (X,Y, δω, γ)

with ∂ the connecting morphism for relative singular homology and δ theconnecting morphism for relative de Rham cohomology.

241

242CHAPTER 12. FORMAL PERIODS AND THE PERIOD CONJECTURE

We write [X,D, ω, γ] for the image of the generator. The vector space Peff(k) isturned into an algebra via

(X,D, ω, γ)(X ′, D′, ω′, γ′) = (X ×X ′, D ×X ′ ∪D′ ×X,ω ∧ ω′, γ × γ′) .

The space of formal periods is the localization P(k) of Peff(k) with respect to[Gm, 1, dXX , S1], where S1 is the unit circle in C∗.

Remark 12.1.2. This is modeled after Kontsevich [K1] Definition 20, but doesnot agree with it. We will discuss this point in more detail in Remark 12.1.7.

Theorem 12.1.3. (Nori) Let k ⊂ C be subfield. Let Gmot(k) be the Tan-nakian dual of the category of Nori motives with Q-coefficents (sic!), see Defini-tion 8.1.6. Let X = SpecP(k). Then X is naturally isomorphic to the torsor ofisomorphisms between singular cohomology and algebraic de Rham cohomologyon Nori motives. It has a natural torsor structure under the base change ofGmot(k,Q) to k (in the fpqc-topology on the category of k-schemes):

X ×k Gmot(k,Q)k → X.

Remark 12.1.4. This was first formulated in the case k = Q without proof byKontsevich as [K1, Theorem 6]. He attributes it to Nori.

Proof. Consider the diagram Pairseff of Definition 8.1.1 and the representationsT1 = H∗dR(−) and T2 = H∗(−, k) (sic!). Note that Hd(X(C), D(C),Q) is dualto Hd(X(C), D(C),Q).

By the very definition, Peff(k) is the module P1,2(Pairseff) of Definition 7.4.19.

By Theorem 7.4.21, it agrees with the module A1,2(Pairseff) of Definition 7.4.2.We are now in the situation of Section 7.4 and apply its main result, Theorem7.4.10. In particular,

A1,2(Pairseff) = A1,2(MMeffNori).

Recall that by Theorem 8.2.20, the diagram categories of Pairseff and Goodeff

agree. This also shows that the modules

A1,2(Pairseff) = A1,2(Goodeff)

agree. From now on, we may work with the diagram Goodeff which has the ad-vantage of admitting a commutative product structure. The algebra structureson A1,2(Goodeff) = P1,2(Goodeff) = Peff(k) agree.

We can apply the same considerations to the localized diagram Good. As inProposition 7.2.5, localization on the level of diagrams or categories amounts tolocalization on the algebra. Hence,

A1,2(Good) = P1,2(Good) = P(k)

andX = SpecA1,2(Good).

12.1. FORMAL PERIODS AND NORI MOTIVES 243

Also, by definition, G2(Good) is the Tannakian dual of the category of Norimotives with k coefficients. By base change Lemma 6.5.6 it is the base changeof the Tannaka dual of the category of Nori motives with Q-coefficients. Afterthese identifications, the operation

X ×k Gmot(k,Q)k → X

is the one of Theorem 7.4.7.

By Theorem 7.4.10, it is a torsor because MMNori is rigid.

Remark 12.1.5. There is a small subtlety here because our to fibre functorstake values in different categories, Q−Mod and k−Mod. As H∗(X,Y, k) =H∗(X,Y,Q)⊗Q k and P(k) already is a k-algebra, the algebra of formal periodsdoes not change when replacing Q-coefficients with k-coefficients.

We can also view X as torsor in the sense of Definition 1.7.9. The descriptionof the torsor structure was discussed extensively in Section 7.4, in particularTheorem 7.4.10. In terms of period matrices, it is given by the formula in [K1]:

Pij 7→∑k,`

Pik ⊗ P−1k` ⊗ P`j .

Corollary 12.1.6. 1. The algebra of effective formal periods Peff(k) remainsunchanged when we restrict in Definition 12.1.1 to (X,D, ω, γ) with Xaffine of dimension d, D of dimension d − 1 and X r D smooth, ω ∈Hd

dR(X,D), γ ∈ Hd(X(C), D(C),Q).

2. Peff(k) is generated as Q-vector space by elements of the form [X,D, ω, γ]with X smooth of dimension d, D a divisor with normal crossings ω ∈Hd

dR(X,D), γ ∈ Hd(X(C), D(C),Q).

Proof. In the proof of Theorem 12.1.3, we have already argued that we canreplace the diagram Pairseff by the diagram Goodeff . The same argument alsoallows to replace it by VGoodeff .

By blowing up X, we get another good pair (X, D, d). By excision, they have thesame de Rham and singular cohomology as (X,D, d). Hence, we may identifythe generators.

Remark 12.1.7. We do not know whether it is enough to work only withformal periods of the form (X,D, ω, γ) with X smooth and D a divisor withnormal crossings in Definition 12.1.1 as Kontsevich does in [K1, Definition 20].By the Corollary, these symbols generate the algebra, but it is not clear to usif they also give all relations. Indeed, Kontsevich in loc. cit. only imposes therelation given by the connecting morphism of triples in an even more specialcase.

Moreover, Kontsevich considers differential forms of top degree rather than co-homology classes. They are automatically closed. He imposes Stokes’ formula


as an additional relation, hence this amounts to considering cohomology classes.Note, however, that not every de Rham class is of this form in general.

All formal effective periods (X,D, ω, γ) can be evaluated by ”integrating” ωalong γ. More precisely, recall (see Definition 5.4.1 the period pairing

HddR(X,D)×Hd(X(C), D(C))→ C

It maps (Gm, 1, dX/X, S1) to 2πi.

Definition 12.1.8. Letev : P(k)→ C,

be the ring homomorphism induced by the period pairing. We denote by perthe C-valued point of X = Spec P(k) defined by ev.

The elements in the image are precisely the element of the period algebra P(k)of Definition 9.3.1. By the results in Chapters 9, 10, and 11 (for k = Q), itagrees with all other definitions of a period algebra. From this perspective, peris the C-valued point of the torsor X of Theorem 12.1.3 comparing singularand algebraic de Rham cohomology. It is given by the period isomorphism perdefined in Chapter 5.

The following statement of period number is a corollary from our previous resultson formal periods.

Corollary 12.1.9. The algebra P(k) is Q-linearly generated by number of theform (2πi)jα with j ∈ Z, and α the period of (X,D, ω, γ) with X smooth affine,D a divisor with normal crossings, ω ∈ ΩdX(X).

This was also proved without mentioning motives as Theorem 9.4.2.

Proof. Recall that 2πi is itself a period of such a quadruple.

By Corollary 8.2.21, the category MMeffNori is generated by motives of good

pairs (X,Y, d) of the form X = W \W∞, Y = W0 \ (W∞ ∩W0) with W smoothprojective of dimension d, W0 ∪W∞ a divisor with normal crossings, X ′ = \W0

affine. Hence, their periods generated Peff(k) as a Q-vector space.

Let Y ′ = W∞ \ (W0 ∩W∞). By Lemma 8.3.8, the motive HdNori(X,Y ) is dual

to HdNori(X

′, Y ′)(d). By Lemma 9.2.9, this implies that the periods of the firstagree with the periods of the latter up to a factor (2πi)d.

As X ′ is affine and Y ′ a divisor with normal crossings, HddR(X ′, Y ′) is generated

by ΩdX′(X′) by Proposition 3.3.19.

Proposition 12.1.10. Let K/k be algebraic. Then

P(K) = P(k) ,

and hence alsoP(K) = P(k) .

12.2. THE PERIOD CONJECTURE 245

The second statement was already proved directly Corollary 9.3.5

Proof. It suffices to consider the case K/k finite. The general case follows bytaking limits.

Generators of P(k) also define generators of P(K) by base change for the fieldextension K/k. The same is true for relations, hence we get a well-defined mapP(k)→ P(K).

We define a map in the opposite direction by viewing a K-variety as k-variety.More precisely, let (Y,E,m) be vertex of Pairseff(K) and (Yk, Ek,m) the sameviewed as vertex of Pairseff(k). As in the proof of Corollary 9.3.5, we have

H(Yk, Ek,m) = RK/kH(Y,E,m)

with RK/k as defined in Lemma 9.2.7. The same proof as in Lemma 9.2.7(treating actual periods) also shows that the formal periods of (Yk, Ek,m) agreewith the formal periods (Y,E,m):

12.2 The period conjecture

We exlore the relation to transcendence questions from the point of view of ofNori motives and their periods. We only treat the case where k/Q is algebraic.For more general fields, see Ayoub’s remarks in [Ay].

Recall that P(Q) = P(k) = P(Q) under this assumption.

Conjecture 12.2.1 (Kontsevich-Zagier). Let k/Q be an algebraic field exten-sion contained in C. The evaluation map (see Definition 12.1.8)

ev : P(k)→ P(k)

is bijective.

Remark 12.2.2. We have already seen that the map is surjective. Henceinjectivity is the true issue. Equivalently, we can conjecture that P(k) is anintegral domain and ev a generic point.

In the literature [A1, A2, Ay, BC, Wu], there are sometimes alternative for-mulations of this conjecture, called ”Grothendieck conjecture”. We will explainthis a little bit more.

Definition 12.2.3. Let M ∈MMNori be a Nori motive. Let

X(M)

be the torsor of isomorphisms between singular and algebraic de Rham coho-mology on the Tannaka category 〈M,M∨〉⊗ generated by M and

P(M) = O(X(M))


the associated ring of formal periods. If M = H∗Nori(Y ) for a variety Y , we also

write P(Y ).

Let Gmot(M) and Gmot(Y ) be the Tannaka duals of the category with respectto singular cohomology.

These are the finite dimensional building blocks of P(k) and Gmot(k), respec-tively.

Remark 12.2.4. By Theorem 7.4.10, the space X(M) is a Gmot(M)-torsor.Hence they share all properties that can be tested after a faithfully flat basechange. In particular, they have the same dimension. Moreover, X(M) issmooth because G(M) is a group scheme over a field of characteristic zero.

Analogous to [Ay] and [A2, Prop. 7.5.2.2 and Prop. 23.1.4.1], we can ask:

Conjecture 12.2.5 (Grothendieck conjecture for Nori motives). Let k/Q bean algebraic extension contained in C and M ∈ MMNori(k). The followingequivalent assertions are true:

1. The evaluation mapev : P(M)→ C

is injective.

2. The point evM of Spec P(M) is a generic point, and X(M) connected.

3. The space X(M) is connected, and the transcendence degree of the sub-field of C generated by the image of evM is the same as the dimension ofGmot(M).

Proof of equivalence. Assume that ev is injective. Then P(M) is contained inthe field C, hence integral. The map to C factors via the residue field of apoint. It ev is injective, this has to be the generic point. The subfield generatedby ev(M) is isomorphic to the function field. Its transcendence degree is thedimension of the integral domain.

Conversely, if X(M) is connected, then it P(M) is integral because it is alreadysmooth. If ev factors the generic point, its function field embeds into C andhence P(M) does. If the subfield generated by the image of ev in C has themaximal possible transcendence degree, then ev has to be generic.

Lemma 12.2.6. If Conjecture 12.2.5 is true for all M , then Conjecture 12.2.1holds.

Proof. By construction, we have

P(k) = colimM P(M).

Injectivity of the evaluation maps on the level of every M implies injectivity ofthe transition maps and injectivity of ev on the union.

12.2. THE PERIOD CONJECTURE 247

Remark 12.2.7. The converse is not obvious. It amounts to asking whetherP(M) is contained in P(k). In our description with generators and relations, thismeans that all relations are given by relations within the category 〈M,M∨〉⊗.This is not clear a priori. We have a conditional result in the pure case.

Proposition 12.2.8. Assume that the Hodge conjecture holds for all varieties.Let M be a pure Nori motive. Then P(M) injects into P(k).

Proof. The algebra P(M) is generated by classes (ω, γ) with ω ∈ H∗dR(M) ⊕H∗dR(M)∨ and γ ∈ H∗(M,Q) ⊕H∗(M,Q)∨ of the same cohomological degree.The relations are given by chains of morphisms and morphisms in the oppositedirection

M →M1 ←M2 → · · · ←M

in the tensor category generated by the direct sum of these Nori motives.

In P(k), the relations between these same generators are given by chains in thecategory of all Nori motives. A priori, there are more of these.

By Proposition 10.4.5, we have a weight filtration on the category of Nori mo-tives. Morphisms between pure motives of different weights vanish. We chooseour generators pure and we apply the weight filtration to the whole chain defin-ing a relation. This implies that there are no relations between pure generatorsof different weights. The relations between pure generators of the same weightare already induced from relations of this fixed weight. We now apply the Hodgeconjecture again and in a semi-simple category. The only relations are the onesgiven by the simple objects in the subcategory.

The third version of Conjecture 12.2.5 is very close to the point of view takenoriginally by Grothendieck in the pure case. In order to understand the preciserelation, we have to establish some properties first.

We specialize to the case P(Y ) for Y smooth and projective. In this case,singular cohomology H∗(Y,Q) carries a pure Q-Hodge structure, see Defini-tion 10.2.2. Recall that the Mumford-Tate group MT(V ) of a polarizable pureHodge structure V is the smallest Q-algebraic subgroup of GL(V ) such thatHodge representation h : S → GL(VR) factors via G as h : S → GR. Here,S = ResC/RGm is the Deligne torus. It is precisely the Q-algebraic subgroupof GL(VR) that fixes all Hodge tensors in all tensor powers

⊕V ⊗m ⊗ V ∨⊗n

[M]. Alternatively, it can be understood as the Tannaka dual of the subcate-gory of the category of Hodge structures generated by V . The group MT(V ) isa reductive Q-algebraic group by [GGK, Chap. I].

Proposition 12.2.9. Let k = Q and let Y be smooth and projective. Assumethat the Hodge conjecture holds for all powers of Y . Then Gmot(Y ) is the sameas the Mumford-Tate group of Y .

Proof. By Proposition 10.4.4 the Tannaka subcategory ofMMNori generated byH∗Nori(Y ) agrees with the Tannaka subcategory of the category of Grothendieck


motives GRM. Note that the statement of Proposition 10.4.4 assumes the fullHodge conjecture. The same argument also gives the statement on the subcat-egories under the weaker assumption. For the rest of the argument we referto Lemme 7.2.2.1 and Remarque 23.1.4.2 of [A2]. It amounts to saying thatequivalent Tannaka categories have isomorphic Tannaka duals.

Corollary 12.2.10 (Period Conjecture). Let Y be a smooth, projective varietyover Q. Assume Conjecture 12.2.5 for powers of Y and the Hodge conjecture.Then every polynomial relation among the periods of Y are of motivic nature,i.e., they are induced by algebraic cycles (correspondences) in powers of Y .

In the case of elliptic curves this was stated as conjecture by Grothendieck[Gro1].

Proof. By Conjecture 12.2.5 all Q-linear relations between periods are inducedby morphisms of Nori motives. Under the Hodge conjecture, the category ofpure Nori motives is equivalent to the category of Grothendieck motives byProposition 10.4.4. By definition of Grothendieck motives (Definition 10.4.2)this means that morphisms are induced from algebraic cycles.

Polynomial relations are induced from the tensor structure, hence powers ofY .

Arnold [Ar, pg. 93] remarked in a footnote that this is related to a conjectureof Leibniz which he made in a letter to Huygens from 1691. Leibniz essentiallyclaims that all periods of generic meromorphic 1-forms are transcendental. Ofcourse, precisely the meaning of ”generic” is the essential question. The conjec-ture of Leibniz can be rephrased in modern form as in [Wu]:

Conjecture 12.2.11 (Integral Conjecture of Leibniz). Any period integral ofa rational algebraic 1-form ω on a smooth projective variety X over a numberfield k over a path γ with ∂γ ⊂ D (the polar divisor of ω) which does not comefrom a proper mixed k-Hodge substructure H ⊆ H1(X \D) is transcendental.

This is only a statement about periods of type i = 1, i.e., for H1(X,D) (or,by duality H1(X \ D)) on curves. The Leibniz conjecture follows essentiallyfrom the period conjecture in the case i = 1, since the Hodge conjecture holdson H1(X) ⊗H1(X) ⊂ H2(X). This conjecture is still open. See also [BC] forstrongly related questions.

Wustholz [Wu] has related this problem to many other transcendance results.One can give transcendance proofs assuming this conjecture:

Example 12.2.12. Let us show that log(α) is transcendental for every algebraicα 6= 0, 1 under the assumption of the Leibniz conjecture. One takes X = P1,and ω = d log(z) and γ = [1, α]. The polar divisor of ω is D = 0,∞, andthe Hodge structure H1(X \ D) = H1(C×) = Z(1) is irreducible as a Hodgestructure. Hence, log(α) is transcendental assuming Leibniz’s conjecture.

12.3. THE CASE OF 0-DIMENSIONAL VARIETIES 249

There are also examples of elliptic curves in [Wu] related to Chudnovsky’s the-orem we mention below.

The third form of Conjecture 12.2.5 is also very useful in a computational sense.In this case, assuming the Hodge conjecture for all powers of Y , the motivic Ga-lois group Gmot(Y ) is the same as the Mumford-Tate group MT(Y ) by Propo-sition 12.2.9.

Andre shows in [A2, Rem. 23.1.4.2]:

Corollary 12.2.13. Let Y be a smooth, projective variety over Q and as-sume that the Hodge conjecture holds for all powers of Y . Then, assumingGrothendieck’s conjecture,

trdegQP(Y ) = dimQ MT(Y ).

Proof. We view the right hand side as Gmot(YQ) by Proposition 12.2.9. By [A2,Paragraph 7.6.4], it is of finite index in Gmot(Y ), hence has the same dimen-sion. It has also the same dimension as the torsor P(Y ). Under Grothendieck’sconjecture, this is given by the transcendence degree of P(Y ), see Conjecture12.2.5.

This corollary give a reasonable, completely unconditional testing conjecture fortranscendence questions.

Example 12.2.14. (Tate motives) If the motive of Y is a Tate motives, e.g.,Y = Pn, then the conjecture is true, since 2πi is transcendant. The Mumford-Tate group is the 1-torus here. More generally, the conjecture holds for Artin-Tate motives, since the transcendance degree remains 1.

Example 12.2.15. (Elliptic curves) Let E be an elliptic curve over Q. Then theMumford-Tate group of E is either a 2-torus if E has complex multiplication, orGL2,Q otherwise (see [M]). Hence, the transcendence degree of P(E) is either 2or 4. G. V. Chudnovsky [Ch] has proved that trdegQP(E) = 2 if E is an ellipticcurve with complex multiplication, and it is ≥ 2 for all elliptic curves over Q.Note that in this situation we have actually 5 period numbers ω1, ω2, η1, η2 andπ around (see Section 13.4 for more details), but they are related by Legendre’srelation ω2η1−ω1η2 = 2πi, so that the transcendence degree cannot go beyond4. Hence, it remains to show that the transcendence degree of the periods of anelliptic curve without complex multiplication is precisely 4, as predicted by theconjecture.

12.3 The case of 0-dimensional varieties

We go through all objects in the baby case of zero motives, i.e., the ones gener-ated by 0-dimensional varieties.


Definition 12.3.1. Let Pairs0 ⊂ Pairseff be the subdiagram of vertices (X,Y, n)with dimX = 0. LetMM0

Nori be its diagram category with respect to the repre-sentation of Pairseff given by singular cohomology with rational coefficients. LetVar0 ⊂ Pairs0 be the diagram defined by the opposite category of 0-dimensionalk-varieties, or equivalently, the category of finite separable k-algebras.

If dimX = 0, then dimY = 0 and X decomposes into a disjoint union of Yand X \ Y . Hence H∗(X,Y,Q) = H∗(X \ Y,Q) and it suffices to consideronly vertices with Y = ∅. Moreover, all cohomology is concentrated in degree0, and the pairs (X,Y, 0) are all good and even very good. In particular, themultiplicative structure on Good restricts to the obvious multiplicative structureon Pairs0 and Var0.

We are always going to work with the multiplicative diagram Var0 in the sequel.

Definition 12.3.2. Let G0mot(k) be the Tannaka dual of MM0

Nori and P0(k)be the space of periods attached to MM0

Nori.

The notation is a bit awkward because G0 often denotes the connected compo-nent of unity of a group scheme G. Our G0

mot(k) is very much not connected.

Our aim is to show that G0mot(k) = Gal(k/k) and P0(k) ∼= k with the natural

operation. In particular, the period conjecture (in any version) holds for 0-motives. This is essentially Grothendieck’s treatment of Galois theory.

By construction of the coalgebra in Corollary 6.5.5, we have

A(Var0, H0) = colimFEnd(H0|F )∨ ,

where F runs through a system of finite subdiagrams whose union is D.

We start with the case when F has a single vertex SpecK, with K/k be a finitefield extension, Y = SpecK. The endomorphisms of the vertex are given by theelements of the Galois group G = Gal(K/k). We spell out H0(Y,Q). We have

Y (C) = Mork(SpecC,SpecK) = Homk−alg(K,C)

the set of field embeddings of K into C, viewed as a finite set with the discretetopology. Singular cohomology attaches a copy of Q to each point, hence

H0(Y (C),Q) = Maps(Y (C),Q) = Maps(Homk−alg(K,C),Q).

As always, this is contravariant in Y , hence covariant in fields. The left operationof the Galois group G on K induces a left operation on H0(Y (C),Q).

Let K/k be Galois of degree d. We compute the ring of endomorphisms of H0

on the single vertex SpecK (see Definition 6.1.8)

E = End(H0|SpecK).

By definition, these are the endomorphisms of H0(SpecK,Q) commuting withthe operation of the Galois group. The set Y (C) has a simply transitive action of

12.3. THE CASE OF 0-DIMENSIONAL VARIETIES 251

G. Hence, Maps(Y (C),Q) is a free Q[G]op-module of rank 1. Its commutator Eis then isomorphic to Q[G]. This statement already makes the algebra structureon E explicit.

The diagram algebra does not change when we consider the diagram Var0(K)containing all vertices of the form A with A =

⊕ni=1Ki, Ki ⊂ K.

There are two essential cases: If K ′ ⊂ K is a subfield, we have a surjective mapY (C) → Y ′(C). The compatibility condition with respect to this map impliesthat the value of the diagram endomorphism on K ′ is already determined by itsvalue on K. If A = K

⊕K, then compatibility with the inclusion of the first

and the second factor implies that the value of the diagram endomorphism onA is already determined by its value on K.

In more abstract language: The category Var0(K) is equivalent to the categoryof finite G-sets. The algebra E is the group ring of the Galois group of thiscategory under the representation S 7→ Maps(S,Q).

Note that K ⊗k K =⊕

σK, with σ running through the Galois group, is inVar0(K). The category has fibre products. In the language of Definition 7.1.3,the diagram Var0(K) has a commutative product structure (with trivial grad-ing). By Proposition 7.1.5 and its proof, the diagram category is a tensorcategory, or equivalently, E carries a comultiplication.

We go through the construction in the proof of loc.cit. We start with an elementof E and view it as an endomorphism of H0(Y × Y (C),Q) ∼= H0(Y (C),Q) ⊗H0(Y (C),Q), hence as a tensor product of endomorphisms of H0(Y (C),Q). Theoperation of E = Q[G] on Maps(Y (C)×Y (C),Q) is determined by the conditionthat it has to be compatible with the diagonal map Y (C)→ Y (C)×Y (C). Thisamounts to the diagonal embedding Q[G]→ Q[G]⊗Q[G].

Thus we have shown that E = Q[G] as bialgebra. This means that

Gmot(Y ) = SpecE∨ = G

as a constant monoid (even group) scheme over Q.

Passing to the limit over all K we get

G0mot(k) = Gal(k/k)

as proalgebraic group schemes of dimension 0. As a byproduct, we see that themonoid attachted to MM0

Nori is a group, hence the category is rigid.

We now turn to periods, again in the case K/k finite and Galois. Note thatH0

dR(SpecK) = K and the period isomorphism

K ⊗k C→ Maps(Homk−alg(K,C),Q)⊗Q C,v 7→ (f 7→ f(v))

is the base change of the same map with values in K

K ⊗k K → Maps(Homk−alg(K,K),Q)⊗Q K.


In particular, all entries of the period matrix are in K. The space of formalperiods of K is generated by symbols (ω, γ) where ω runs through a k-basis ofK and γ through the set Homk−alg(K,K) viewed as basis of a Q-vector space.The relations coming from the operation of Galois group bring us down to aspace of dimension [K : k], hence the evaluation map is injective. Passing tothe limit, we get

P0(k) = k.

(We would get the same result by applying Proposition 12.1.10 and working onlyover k.) The operation of Gal(k/k) on P0(k) is the natural one. More precisely,g ∈ Gal(k/k) operates by applying g−1 because the operation is defined via γ,which is in the dual space. Note that the dimension of P0(k) is also 0.

We have seen from general principles that the operation of Gal(k/k) on X0(k) =P0(k) defines a torsor. In this case, we can trivialize it already over k. We have

Mork(Speck, X0(k)) = Homk−alg(k, k).

By Galois theory, the operation of Gal(k/k) on this set is simply transitive.

When we apply the same discussion to the ground field k, we get G0mot(k) =

Gal(k/k) and P0(k) = k. We see that the (formal) period algebra has notchanged, but the motivic Galois group has. It is still true that Speck is atorsor under the motivic Galois group, but now viewed as k-schemes, whereboth consist of a single point!

Part IV

Examples

253

Chapter 13

Elementary examples

13.1 Logarithms

In this section, we give one of the most simple examples for a cohomologicalperiod in the sense of Chap. 9. Let

X := A1Q \ 0 = SpecQ[t, t−1]

be the affine line with the point 0 deleted and

D := 1, α with α 6= 0, 1

a divisor on X. The singular homology of the pair (X(C), D(C)) = (C×, 1, α)is generated by a small loop σ turning counter-clockwise around 0 once andthe interval [1, α]. In order to compute the algebraic de Rham cohomology of(X,D), we first note that by Section 3.2, H•dR(X,D) is the cohomology of the

complex of global sections of the cone complex Ω•X,D, since X is affine and the

sheaves ΩpX,D are quasi-coherent, hence acyclic for the global section functor.

We spell out the complex Γ(X, Ω•X,D) in detail

0xΓ(X, Ω1

X,D) = Γ(X,Ω1

X ⊕⊕j

i∗ODj)

= Q[t, t−1]dt⊕Q1⊕Q

αxdΓ(X,OX) = Q[t, t−1]

255

256 CHAPTER 13. ELEMENTARY EXAMPLES

and observe that the evaluation map

Q[t, t−1] Q1⊕Q

α

f(t) 7→(f(1), f(α)

)is surjective with kernel

(t− 1)(t− α)Q[t, t−1] = spanQtn+2 − (α+ 1)tn+1 + αtn |n ∈ Z.

Differentiation maps this kernel to

spanQ(n+ 2)tn+1 − (n+ 1)(α+ 1)tn − nαtn−1 |n ∈ Zdt.

Therefore we get

H1dR(X,D) = Γ(X0, ΩX,D) /Γ(X,OX)

= Q[t, t−1]dt⊕Q1⊕Q

α/ d(Q[t, t−1])

= Q[t, t−1]dt/ spanQ(n+ 2)tn+1 − (n+ 1)(α+ 1)tn − nαtn−1dt.

By the last line, we see that the class of tndt in H1dR(X,D) for n 6= −1 is linearly

dependent of

• tn−1dt and tn−2dt, and

• tn+1dt and tn+2dt,

hence we see by induction that dtt and dt generate H1

dR(X,D). Therefore,H1

dR(X,D) is spanned by

dt

tand

1

α− 1dt.

We obtain the following period matrix P for H1(X,D):

1α−1dt

dtt

[1, α] 1 logα

σ 0 2πi

(13.1)

In Section 7.4.3 we have seen how the torsor structure on the periods of (X,D)is given by a triple coproduct ∆ in terms of the matrix P :

Pij 7→∑k,`

Pik ⊗ P−1k` ⊗ P`j .

The inverse period matrix in this example is given by:

P−1 =

(1 − logα

2πi

0 12πi

)and thus we get for the triple coproduct of the most important entry log(α)

∆(logα) = logα⊗ 12πi ⊗ 2πi− 1⊗ logα

2πi ⊗ 2πi+ 1⊗ 1⊗ logα . (13.2)

13.2. MORE LOGARITHMS 257

13.2 More Logarithms

In this section, we describe a variant of the cohomological period in the previoussection. We define

D0 := 1, α, β with α 6= 0, 1 and β 6= 0, 1, α,

but keep X := A1Q \ 0 = SpecQ[t, t−1].

Then, Hsing1 (X,D;Q) is generated by the loop σ from the first example and the

intervals [1, α] and [α, β]. Hence, the differential forms dtt , dt and 2t dt give a

basis of H1dR(X,D): If they were linearly dependent, the period matrix P would

not be of full rank

dtt dt 2t dt

σ 2πi 0 0

[1, α] logα α− 1 α2 − 1

[α, β] log(βα

)β − α β2 − α2 .

We observe that detP = 2πi(α− 1)(β − α)(β − 1) 6= 0.

We have

P−1 =

1

2πi 0 0

log β(α2−1)−logα(β2−1)2πi(β−α)(α−1)(β−1)

α+β(α−1)(β−1)

α+1(α−β)(β−1)

− log β(α−1)+logα(β−1)2πi(β−α)(α−1)(β−1)

−1(α−1)(β−1)

−1(α−β)(β−1)

,

and therefore we get for the triple coproduct for the entry log(α):

∆(logα) = logα⊗ 1

2πi⊗ 2πi

+ (α− 1)⊗ − log β(α2 − 1) + logα(β2 − 1)

2πi(β − α)(α− 1)(β − 1)⊗ 2πi

+ (α− 1)⊗ α+ β

(α− 1)(β − 1)⊗ logα

+ (α− 1)⊗ α+ 1

(α− β)(β − 1)⊗ log

(β

α

)+ (α2 − 1)⊗ log β(α− 1)− logα(β − 1)

2πi(β − α)(α− 1)(β − 1)⊗ 2πi

+ (α2 − 1)⊗ −1

(α− 1)(β − 1)⊗ logα

+ (α2 − 1)⊗ −1

(α− β)(β − 1)⊗ log

(β

α

)= logα⊗ 1

2πi⊗ 2πi− 1⊗ logα

2πi⊗ 2πi+ 1⊗ 1⊗ logα .

Compare this with Equation 13.2 !


13.3 Quadratic Forms

LetQ(x) : Q3 −→ Q

x = (x0, x1, x2) 7→ xAxT

be a quadratic form with A ∈ Q3×3 being a regular, symmetric matrix.

The zero-locus of Q(x)

X := x ∈ P2(Q) |Q(x) = 0

is a quadric or non-degenerate conic. We are interested in its affine piece

X := X ∩ x0 6= 0 ⊂ Q2 ⊂ P2(Q).

We show that we can assume Q(x) to be of a particular nice form. A non-zerovector v ∈ Q3 is called Q-anisotropic, if Q(v) 6= 0. Since charQ 6= 2, there existsuch vectors, just suppose the contrary:

Q(1, 0, 0) = 0 gives A11 = 0,

Q(0, 1, 0) = 0 gives A22 = 0,

Q(1, 1, 0) = 0 gives 2 ·A12 = 0

and A would be degenerate. In particular

Q(1, λ, 0) = Q(1, 0, 0) + 2λQ(1, 1, 0) + λ2Q(0, 1, 0)

will be different form zero for almost all λ ∈ Q. Hence, we can assume that(1, 0, 0) is anisotropic after applying a coordinate transformation of the form

x′0 := x0, x′1 := −λx0 + x1, x′2 := x2.

After another affine change of coordinates, we can also assume that A is adiagonal matrix. An inspection reveals that we can choose this coordinatetransformation such that the x0-coordinate is left unaltered. (Just take for e1

the anisotropic vector (1, 0, 0) in the proof.) Such a transformation does notchange the isomorphism type of X, and we can take X to be cut out by anequation of the form

ax2 + by2 = 1 for a, b ∈ Q×

with affine coordinates x := x1

x0and y := x2

x0. Since X is affine, the sheaves ΩpX

are acyclic, hence we can compute its algebraic de Rham cohomology by

H•dR(X) = h•Γ(X,Ω•X),

13.3. QUADRATIC FORMS 259

so we write down the complex Γ(X,Ω•X) in detail

0

↑Γ(X,Ω1

X) = Q[x, y]/(ax2 + by2 − 1)dx, dy / (axdx+ bydy)

d ↑Γ(X,OX) = Q[x, y]/(ax2 + by2 − 1).

Obviously, H1dR(X) can be presented with generators xnymdx and xnymdy for

m,n ∈ N0 modulo numerous relations. Using axdx+ bydy = 0, we get

• ym dy = d ym+1

m+1 ∼ 0

• xn dx = d xn+1

n+1 ∼ 0

n ≥ 1 • xnym dy = −nm+1x

n−1ym+1 dx+ d xnym+1

m+1

∼ −nm+1x

n−1ym+1 dx for n ≥ 1,m ≥ 0

• xny2m dx = xn(

1−ax2

b

)mdx ∼ 0

• xny2m+1 dx = xn(

1−ax2

b

)my dx

• xy dx = −x2

2 dy + d x2y2

∼ by2−12a dy

= b2ay

2 dy − 12a dy ∼ 0

n ≥ 2 • xny dx = −ba x

n−1y2 dy + xny dx+ bax

n−1y2 dy

= −ba x

n−1y2 dy + xn−1y2a d(ax2 + by2 − 1)

= −ba x

n−1y2 dy + d( (xn−1y)(ax2+by2−1)

2a

)∼ −ba x

n−1y2 dy

=(xn+1 − xn−1

a

)dy

=(− (n+ 1)xny + n−1

a xn−2y)dx+ d

(xn+1y − xn−1

a y)

⇒ xny dx ∼ n−1(n+2)ax

n−2y dx for n ≥ 2.

Thus we see that all generators are linearly dependent of y dx

H1dR(X) = h1Γ(X,Ω•X) = Q y dx.

What about the base change to C of X? We use the symbol√

for the principalbranch of the square root. Over C, the change of coordinates

u :=√ax− i

√by, v :=

√ax+ i

√by


gives

X = SpecC[x, y]/(ax2 + by2 − 1)

= SpecC[u, v]/(uv − 1)

= SpecC[u, u−1]

=A1C \ 0.

Hence, the first singular homology group Hsing• (X,Q) of X is generated by

σ : [0, 1]→ X(C), s 7→ u = e2πis,

i.e., a circle with radius 1 turning counter-clockwise around u = 0 once.

The period matrix consists of a single entry∫σ

y dx =

∫σ

v − u2i√bdu+ v

2√a

Stokes=

∫σ

v du− u dv4i√ab

=1

2i√ab

∫σ

du

u

=π√ab.

The denominator squared is nothing but the discriminant of the quadratic formQ

discQ := detA ∈ Q×/Q×2.

This is an important invariant, that distinguishes some, but not all isomorphismclasses of quadratic forms. Since discQ is well-defined modulo (Q×)2, it makessense to write

H1dR(X) = Q

π√discQ

⊂ H1sing(X,Q)⊗Q C.

13.4 Elliptic Curves

In this section, we give another well-known example for a cohomological periodin the sense of Chap. 9.

An elliptic curve E is a one-dimensional non-singular complete and connectedgroup variety over a field k, together with the origin 0, a k-rational point. Anelliptic curve has genus g = 1, where the genus g of a smooth projective curveis defined as

g := dimk Γ(E,Ω1E) .

We refer to the book [Sil] of Silverman for the theory of elliptic curves, but tryto be self-contained in the following. For simplicity, we assume k = Q. It can

13.4. ELLIPTIC CURVES 261

be shown, using the Riemann-Roch theorem that such an elliptic curve E canbe given as the zero locus in P2(Q) of a Weierstraß equation

Y 2Z = 4X3 − g2XZ2 − g3Z

3 (13.3)

with Eisenstein series coefficients g2 = 60G4, g3 = 140G6 and projective coordi-nates X, Y and Z.

By the classification of compact, oriented real surfaces, the base change of E toC gives us a complex torus Ean, i.e., an isomorphism

Ean ∼= C/Λω1, ω2 (13.4)

in the complex analytic category with

Λω1, ω2:= ω1Z⊕ ω2Z

for ω1, ω2 ∈ C linearly independent over R,

being a lattice of full rank. Thus, all elliptic curves over C are diffeomorphicto the standard torus S1 × S1, but carry different complex structures as theparameter τ := ω2/ω1 varies. We can describe the isomorphism (13.4) quiteexplicitly using periods. Let α and β be a basis of

Hsing1 (E,Z) = Hsing

1 (S1 × S1,Z) = Zα ⊕ Zβ.

The Q-vector space Γ(E,Ω1E) is spanned by the holomorphic differential

ω =dX

Y.

The mapEan → C/Λω1, ω2

P 7→∫ P

O

ω modulo Λω1, ω2

(13.5)

then gives the isomorphism of Equation 13.4. Here O = [0 : 1 : 0] denotes thegroup theoretic origin in E. The integrals

ω1 :=

∫α

ω and ω2 :=

∫β

ω

are called the periods of E. Up to a Z-linear change of basis, they are preciselythe above generators of the lattice Λω1, ω2 .

The inverse map C/Λω1, ω2→ Ean for the isomorphism (13.5) can be described

in terms of the Weierstraß ℘-function of the lattice Λ := Λω1, ω2

℘(z) = ℘(z,Λ) :=1

z2+∑ω∈Λω 6=0

1

(z − ω)2− 1

ω2


and takes the form

C/Λω1, ω2→ Ean ⊂ CPan

2

z 7→ [℘(z) : ℘′(z) : 1],Λω1,ω27→ (0 : 1 : 0).

Note that under the natural projection π : C → C/Λω1, ω2 any meromorphicfunction f on the torus C/Λω1, ω2

lifts to a doubly-periodic function π∗f on thecomplex plane C with periods ω1 and ω2

f(x+ nω1 +mω2) = f(x) for all n,m ∈ Z and x ∈ C.

This example is possibly the origin of the “period” terminology.

The defining coefficients G4, G6 of E can be recovered from Λω1, ω2 by the Eisen-stein series

G2k :=∑ω∈Λω 6=0

ω−2k for k = 2, 3.

Therefore, the periods ω1 and ω2 determine the elliptic curve E uniquely. How-ever, they are not invariants of E, since they depend on the chosen Weierstraßequation of E. A change of coordinates which preserves the shape of (13.3),must be of the form

X ′ = u2X, Y ′ = u3Y, Z ′ = Z for u ∈ Q×.

In the new parametrization X ′, Y ′, Z ′, we have

G′4 = u4G4, G′6 = u6G6,

ω′ = u−1ω

ω′1 = u−1ω1 and ω′2 = u−1ω2.

Hence, τ = ω2/ω1 is a better invariant of the isomorphism class of E. The valueof the j-function (a modular function)

j(τ) = 1728g3

2

g32 − 27g2

3

= q−1 + 744 + 196884q + · · · (q = exp(2πiτ)

on τ indeed distinguishes non-isomorphic elliptic curves E over C:

E ∼= E′ if and only if j(E) = j(E′) .

Hence, the moduli space of elliptic curves over C is the affine line.

A similar result holds over any algebraically closed field K of characteristicdifferent from 2, 3. For fields K that are not algebraically closed, the set of K-isomorphism classes of elliptic curves isomorphic over K to a fixed curve E/Kis the Weil-Chatelet group of E [Sil], an infinite group for K a number field.

13.4. ELLIPTIC CURVES 263

However, E has two more cohomological periods which are also called quasi-periods. In section 13.5, we will prove that the meromorphic differential form

η := XdX

Y

spans H1dR(E) together with ω = dX

Y , i.e., modulo exact forms this form is agenerator of H1(E,OE) in the Hodge decomposition. Like ω corresponds to dzunder (13.5), η corresponds to ℘(z)dz. The quasi-periods then are

η1 :=

∫α

η, η2 :=

∫β

η.

We obtain the following period matrix for E:

dXY X dX

Y

α ω1 η1

β ω2 η2

(13.6)

Lemma 13.4.1. One has the Legendre relation (negative determinant of periodmatrix)

ω2η1 − ω1η2 = ±2πi.

Proof. Consider the Weierstraß ζ-function [Sil]

ζ(z) :=1

z+∑ω∈Λω 6=0

(1

z − ω+

1

ω+

z

ω2

).

It satisfies ζ ′(z) = −℘(z). Since ζ ′(z) = −℘(z) and ℘ is periodic, we havethat η(w) = ζ(z + w) − ζ(z) is independent of z. Hence, the complex pathintegral counter-clockwise around the fundamental domain centered at somepoint a /∈ Λω1,ω2 yields

2πi =

∫ a+ω1

a

ζ(z)dz +

∫ a+ω1+ω2

a+ω1

ζ(z)dz −∫ a+ω1+ω2

a+ω2

ζ(z)dz −∫ a+ω2

a

ζ(z)dz

=

∫ a+ω2

a

(ζ(z + ω1)− ζ(z)) dz −∫ a+ω1

a

(ζ(z + ω2)− ζ(z)) dz

= ω2η1 − ω1η2,

where ηi = η(ωi).

In the following two examples, all four periods are calculated and yield Γ-values besides

√π, π and algebraic numbers. Such period expressions for ellip-

tic curves with complex multiplication are nowadays called the Chowla-Lerch-Selberg formula, after Lerch [L] and Chowla-Selberg [CS]. See also the thesis ofB. Gross [Gr].


Example 13.4.2. Let E be the elliptic curve with G6 = 0 and affine equationY 2 = 4X3 − 4X.Then one has [Wa]

ω1 =

∫ ∞1

dx√x3 − x

=1

2B

(1

4,

1

2

)=

Γ(1/4)2

23/2π1/2, ω2 = iω1,

and

η1 =π

ω1=

(2π)3/2

Γ(1/4)2, η2 = −iη1.

E has complex multiplication with ring Z[i] (Gaußian integers).

Example 13.4.3. Look at the elliptic curve E with G4 = 0 and affine equationY 2 = 4X3 − 4.Then one has [Wa]

ω1 =

∫ ∞1

dx√x3 − 1

=1

3B

(1

6,

1

2

)=

Γ(1/3)3

24/3π, ω2 = ρω1,

(ρ = −1+√−3

2 ) and

η1 =2π√3ω1

=27/3π2

31/2Γ(1/3)3, η2 = ρ2η1.

E has complex multiplication with ring Z[ρ] (Eisenstein numbers).

Both of these example have complex multiplication. As we have explained inExample 12.2.15, G. V. Chudnovsky [Ch] has proved that trdegQP(E) = 2 ifE is an elliptic curve with complex multiplication. This means that ω1 andπ are both transcendant and algebraically independent, and ω2, η1 and η2 arealgebraically dependent. The transcendance of ω1 for all elliptic curves is atheorem of Th. Schneider [S]. Of course, the transcendance of π is Lindemann’stheorem.

For elliptic without complex multiplication it is conjectured that the Legendrerelation is the only algebraic relation among the 5 period numbers ω1, ω2, η1,η2 and π. But this is still open.

13.5 Periods of 1-forms on arbitrary curves

Let X be a smooth, projective curve of geometric genus g over k, where k ⊂ C.We denote the associated analytic space by Xan.

In the classical literature, different types of meromorphic differential forms onXan and their periods were considered. The survey of Messing [Me] gives ahistorical account, see also [GH, pg. 459]. In this section, we mention these no-tions, translate them into a modern language, and relate them to cohomologicalperiods in the sense of Chap. 9, since the terminology is still used in many areasof mathematics, e.g., in transcendence theory.

13.5. PERIODS OF 1-FORMS ON ARBITRARY CURVES 265

A meromorphic 1-form ω on Xan is locally given by f(z)dz, where f is meromor-phic. Any meromorphic function has poles in a discrete and finite set D in Xan.Using a local coordinate z at a point P ∈ Xan, we can write f(z) = z−ν(P ) ·h(z),where h is holomorphic and h(P ) 6= 0. In particular, a meromorphic 1-form isa section of the holomorphic line bundle Ω1

Xan(kD) for some integer k ≥ 0. Wesay that ω has logarithmic poles, if ν(P ) ≤ 1 at all points of D. A rational 1-form is a section of the line bundle Ω1

X(kD) on X. In particular, we can speakof rational 1-forms defined over k, if X is defined over k.

Proposition 13.5.1. Meromorphic 1-forms on Xan are the same as rational1-forms on X.

Proof. Since X is projective, and meromorphic 1-forms are section of the linebundle Ω1

X(kD) for some integer k ≥ 0, this follows from Serre’s GAGA principle[Se1].

In the following, we will mostly use the analytic language of meromorphic forms.

Definition 13.5.2. A differential of the first kind on Xan is a holomorphic1-form (hence closed). A differential of the second kind is a closed meromorphic1-form with vanishing residues. A differential of the third kind is a closed mero-morphic 1-form with at most logarithmic poles along some divisor Dan ⊂ Xan.

Note that forms of the second and third kind include forms of the first kind.

Theorem 13.5.3. Any meromorphic 1-form ω on Xan can be written as

ω = df + ω1 + ω2 + ω3,

where df is an exact form, ω1 is of the first kind, ω2 is of the second kind, andω3 is of the third kind. This decomposition is unique up to exact forms, if ω3 ischosen not to be of second kind, and ω2 not to be of the first kind.

The first de Rham cohomology of Xan is given by

H1dR(Xan,C) ∼=

1− forms of the second kind

exact forms

The inclusion of differentials of the first kind into differentials of the secondkind is given by the Hodge filtration

H0(Xan,Ω1Xan) ⊂ H1

dR(Xan,C).

For differentials of the third kind, we note that

F 1H1(Xan rDan,C) = H0(Xan,Ω1Xan〈Dan〉)

∼=1− forms of the third kind with poles in Dan

exact forms + forms of the first kind.


Proof. Let ω be a meromorphic 1-form on Xan. The residue theorem statesthat the sum of the residues of ω is zero. Suppose that ω has poles in the finitesubset D ⊂ Xan. Then look at the exact sequence

0→ H0(Xan,Ω1Xan)→ H0(Xan,Ω1

Xan〈D〉)Res→⊕P∈D

C Σ→H1(Xan,Ω1Xan).

It shows that there exists a 1-form ω3 ∈ H0(Xan,Ω1Xan(logD)) of the third kind

which has the same residues as ω. In addition, the form ω−ω3 is of the secondkind, i.e., it has perhaps poles but no residues. Now, look at the meromorphicde Rham complex

Ω0Xan(∗) d−→Ω1

Xan(∗)

of all meromorphic differential forms on Xan (with arbitrary poles along arbi-trary divisors). The cohomology sheaves of it are given by [GH, pg. 457]

H0Ω•Xan(∗) = C, H1Ω•Xan(∗) =⊕

P∈Xan

C .

These isomorphisms are induced by the inclusion of constant functions and theresidue map respectively. With the help of the spectral sequence abutting toH∗(Xan,Ω∗Xan(∗)) [GH, pg. 458], one obtains an exact sequence

0→ H1dR(Xan,C)→ H0(Xan,Ω1

Xan(∗))exact forms

Res−→⊕

P∈Xan

C,

and the claim follows. The identification with F 1H1(Xan rDan,C) is by defi-nition of the Hodge filtration.

Corollary 13.5.4. In the algebraic category, if X is defined over k ⊂ C, wehave that

H1dR(X) ∼=

1− rational forms of the second kind over k

exact forms

We can now define periods of differentials of the first, second, and third kind.

Definition 13.5.5. Periods of the n-th kind (n=1,2,3) in the sense of Defini-tion 9.1.1 are periods of differentials ω of the n-th kind, i.e., integrals∫

γ

ω ,

where γ is a closed path avoiding the poles of D for n = 2 and which is containedin X \D for n = 3.

Usually, in the literature periods of 1-forms of the first kind are called periods,and periods of 1-forms of the second kind and not of the first kind are calledquasi-periods.

13.5. PERIODS OF 1-FORMS ON ARBITRARY CURVES 267

Theorem 13.5.6. Let X be a smooth, projective curve over k as above.

Periods of the second kind (and hence also periods of the first kind) are coho-mological periods in the sense of 9.3.1 of the first cohomology group H1(X).Periods of the third kind with poles along D are periods of the cohomology groupH1(U), where U = X \D.

Every period of any smooth, quasi-projective curve U over k is of the first, secondor third kind on a smooth compacification X of U .

Proof. The first assertion follows from the definition of periods of the n-th kind,since differentials of the n-th kind represent cohomology classes in H1(X) forn = 1, 2 and in H1(X \D) for n = 3. If U is a smooth, quasiprojective curveover k, then we choose a smooth compactification X and the assertion followsfrom the exact sequence

0→ H0(Xan,Ω1Xan)→ H0(Xan,Ω1

Xan〈D〉)Res→⊕P∈D

C Σ→H1(Xan,Ω1Xan).

by Theorem 13.5.3.

Examples 13.5.7. In the elliptic curve case of section 13.4, ω = dXY is 1-form

of the first kind, and η = X dXY a 1-form of the second kind, but not of the first

kind. Some periods (and quasi-periods) of this sort were computed in the twoExamples 13.4.2,13.4.3. For an example of the third kind, look at X = P1 andD = 0,∞ where ω = dz

z is a generator with period 2πi. Compare this withsection 13.1 where also logarithms occur as periods. For periods of differentialsof the third kind on modular and elliptic curves see [Br].

Finally, let X be a smooth, projective curve of genus g defined over Q. Thenthere is a Q-basis ω1, . . . , ωg, η1, . . . , ηg of H1

dR(X), where the ωi are of the firstkind and the ηj of the second kind. One may choose a basis α1, . . . , αg, β1, . . . , βgfor Hsing

1 (Xan,Z), such that after a change of basis over Q, we have∫αjωi = δij

and∫βjηi = δij .

The period matrix is then given by a block matrix:

ω• η•α• I τ ′

β• τ I(13.7)

where, by Riemann’s bilinear relations [GH, pg. 123], τ is a matrix in theSiegel upper half space Hg of symmetric complex matrices with positive definiteimaginary part. In the example of elliptic curves, section 13.4 the matrix τ isthe (1× 1)-matrix given by τ = ω2/ω1 ∈ H.

For transcendence results for periods of curves and abelian varieties we refer tothe survey of Wustholz [Wu], and our discussion in Section 12.2 of Part III.


Chapter 14

Multiple zeta values

This chapter follows partly the Diploma thesis of Benjamin Friedrich, see [Fr].We study in some detail the very important class of periods called multiple zetavalues (MZV). These are periods of mixed Tate motives.

14.1 A ζ-value

In Prop. 11.1.4, we saw how to write ζ(2) as a Kontsevich-Zagier period:

ζ(2) =

∫0≤ x≤ y≤ 1

dx ∧ dy(1− x) y

.

The problem was that this identity did not give us a valid representation ofζ(2) as a naıve period, since the pole locus of the integrand and the domain ofintegration are not disjoint. We show how to circumvent this difficulty, as anexample of Theorem 11.2.1.

First we define (often ignoring the difference between X and Xan),

Y := A2 with coordinates x and y,

Z := x = 1 ∪ y = 0,X := Y \ Z,D := (x = 0 ∪ y = 1 ∪ x = y) \ Z,4 := (x, y) ∈ Y |x, y ∈ R, 0 ≤ x ≤ y ≤ 1 a triangle in Y, and

ω :=dx ∧ dy(1− x) y

,

thus getting

ζ(2) =

∫4ω,

269

270 CHAPTER 14. MULTIPLE ZETA VALUES

Figure 14.1: The configuration Z,D,4

with ω ∈ Γ(X,Ω2X) and ∂4 ⊂ D ∪ (0, 0), (1, 1), see Figure 14.1.

Now we blow up Y in the points (0, 0) and (1, 1) obtaining π : Y → Y . We

denote the strict transform of Z by Z, π∗ω0 by ω and Y \ Z by X. The “strict

transform” π−1(4 \ (0, 0), (1, 1)) will be called 4 and (being Q-semi-algebraichence triangulable — cf. Proposition 2.6.9) gives rise to a singular chain

γ ∈ Hsing2 (X, D;Q).

Since π is an isomorphism away from the exceptional locus, this exhibits

ζ(2) =

∫4ω =

∫γ

ω ∈ Pa = P

as a naıve period, see Figure 14.2.

Figure 14.2: The configuration Z, D, 4

We will conclude this example by writing out ω and 4 more explicitly. Notethat Y can be described as the subvariety

A2Q × P1(Q)× P1(Q) with coordinates (x, y, λ0 : λ1, µ0 : µ1)

14.2. DEFINITION OF MULTIPLE ZETA VALUES 271

cut out byxλ0 = yλ1 and (x− 1)µ0 = (y − 1)µ1.

With this choice of coordinates π takes the form

π : Y → Y(x, y, λ0 : λ1, µ0 : µ1) 7→ (x, y)

and we have X := Y \ (λ0 = 0∪µ1 = 0). We can embed X into affine space

X → A4Q

(x, y, λ0 : λ1, µ0 : µ1) 7→ (x, y,λ1

λ0,µ0

µ1)

and so have affine coordinates x, y, λ := λ1

λ0and µ := µ0

µ1on X.

Now, near π−1(0, 0), the form ω is given by

ω =dx ∧ dy(1− x) y

=d(λy) ∧ dy(1− x) y

=dλ ∧ dy1− x

,

while near π−1(1, 1) we have

ω =dx ∧ dy(1− x) y

=dx ∧ d(y − 1)

(1− x) y=dx ∧ d(µ(x− 1))

(1− x) y=−dx ∧ dµ

y.

The region 4 is given by

4 = (x, y, λ, µ) ∈ X(C) | x, y, λ, µ ∈ R, 0 ≤ x ≤ y ≤ 1, 0 ≤ λ ≤ 1, 0 ≤ µ ≤ 1.

14.2 Definition of multiple zeta values

Recall that the Riemann ζ-function is defined as

ζ(s) :=

∞∑n=1

n−s, Re(s) > 1.

It has an analytic continuation to the whole complex plane with a simple poleat s = 1.

Definition 14.2.1. For integers s1, ..., sr ≥ 1 with s1 ≥ 2 one defines themultiple zeta values (MZV)

ζ(s1, ..., sr) :=∑

n1>n2>...>nr≥1

n−s11 · · ·n−srr .

The number n = s1 + · · ·+ sr is the weight of ζ(s1, ..., sr). The length is r.


Lemma 14.2.2. ζ(s1, ..., sr) is convergent.

Proof. Clearly, ζ(s1, ..., sr) ≤ ζ(2, 1, ..., 1). We use the formula

m−1∑n=1

n−1 ≤ 1 + log(m− 1),

which is proved by comparing with the Riemann integral of 1/x. Using induc-tion, this implies that

ζ(2, 1, ..., 1) ≤∞∑

n1=1

n−21

∑1≤nr<···<n2≤n1−1

n−12 · · ·n−1

r ≤∞∑

n1=1

(1 + log(n1 − 1))r

n21

,

which is convergent.

Lemma 14.2.3. The positive even ζ-values are given by

ζ(2m) = (−1)m+1 (2π)2m

2(2m)!B2m,

where B2m is a Bernoulli number, defined via

t

et − 1=

∞∑m=0

Bmtm

m!.

The first Bernoulli numbers are B0 = 1, B1 = −1/2, B2 = 1/6, B3 = 0,B4 = −1/30. All odd Bernoulli Bm numbers vanish for odd m ≥ 3.

Proof. One uses the power series

x cot(x) = 1− 2

∞∑n=1

x2

n2π2 − x2.

The geometric series expansion gives

x cot(x) = 1− 2

∞∑n=1

(xnπ

)21−

(xnπ

)2 = 1− 2

∞∑m=1

x2m

π2mζ(2m).

On the other hand,

x cot(x) = ixeix + e−ix

eix − e−ix= ix

e2ix + 1

e2ix − 1= ix+

2ix

e2ix − 1= ix+

∞∑m=0

Bm(2ix)m

m!.

The claim then follows by comparing coefficients.

Corollary 14.2.4. ζ(2) = π2

6 and ζ(4) = π4

90 .

14.2. DEFINITION OF MULTIPLE ZETA VALUES 273

ζ(s) satisfies a functional equation

ζ(s) = 2sπs−1 sin(πs

2

)Γ(1− s)ζ(1− s).

Using it, one can show:

Corollary 14.2.5. ζ(−m) = −Bm+1

m+1 for m ≥ 1. In particular, ζ(−2m) = 0 form ≥ 1. These are called the trivial zeroes of ζ(s).

Remark 14.2.6. J. Zhao has generalized the analytic continuation and thefunctional equation for multiple zeta values [Z2].

In the following, we want to study MZV as periods. They satisfy many relations.Already Euler knew that ζ(2, 1) = ζ(3). This can be shown as follows:

ζ(3) + ζ(2, 1) =

∞∑n=1

1

n3+∑

1≤k<n

1

n2k=

∑1≤k≤n

1

n2k=

∞∑n=1

1

n2

n∑k=1

1

k

=∑k,n≥1

1

n2

(1

k− 1

n+ k

)=∑k,n≥1

1

nk(n+ k)

=∑k,n≥1

(1

n+

1

k

)1

(n+ k)2=∑k,n≥1

1

n(n+ k)2+∑k,n≥1

1

k(n+ k)2

= 2ζ(2, 1).

Other relations of this type are

ζ(2, 1, 1) = ζ(4),

ζ(2, 2) =3

4ζ(4),

ζ(3, 1) =1

4ζ(4),

ζ(2)2 =5

2ζ(4),

ζ(5) = ζ(3, 1, 1) + ζ(2, 1, 2) + ζ(2, 2, 1)

ζ(5) = ζ(4, 1) + ζ(3, 2) + ζ(2, 3).

The last two relations are special cases of the sum relation:

ζ(n) =∑

s1+···+sr=n

ζ(s1, ..., sr).

It was conjectured by Zagier [Z] that the Q-vector space Zn of MZV of weightn has dimension dn, where dn is the coefficient of tn in the power series

∞∑n=0

dntn =

1

1− t2 − t3,


so that one has a recursion dn = dn−2 + dn−3. For example d4 = 1, which canbe checked using the above relations. By convention, d0 = 1. This conjecture isstill open, however one knows that dn is an upper bound for dimQ(Zn) [B1, Te].It is also conjectured that the MZV of different weights are independent overQ, so that the space of all MZV should be a direct sum

Z =⊕n≥0

Zn.

Hoffman [Hof] conjectured that all MZV containing only si ∈ 2, 3 form abasis of Z. Brown [B1] showed in 2010 that this set forms a generating set.Broadhurst et. al. [BBV] conjecture that the ζ(s1, ..., sr) with si ∈ 2, 3 aLyndon word form a transcendence basis. A Lyndon word in two letters withan order, e.g. 2 < 3, is a string in these two letters that is strictly smaller inlexicographic order than all of its circular shifts.

14.3 Kontsevich’s integral representation

Define one-forms ω0 := dtt and ω1 := dt

1−t . We have seen that

ζ(2) =

∫0≤t1≤t2≤1

ω0(t2)ω1(t1).

In a similar way, we get that

ζ(n) =

∫0≤t1≤···≤tn≤1

ω0(tn)ω0(tn−1) · · ·ω1(t1).

We will now write this asζ(n) = I(0 . . . 01︸︷︷︸

n

).

Definition 14.3.1. For ε1, ..., εn ∈ 0, 1, we define the Kontsevich-Zagierperiods

I(εn . . . ε1) :=

∫0≤t1≤···≤tn≤1

ωεn(tn)ωεn−1(tn−1) · · ·ωε1(t1).

Note that this definition differs from parts of the literature in terms of the order,but it has the advantage that there is no sign in the following formula:

Theorem 14.3.2 (Attributed to Kontsevich by Zagier [Z]).

ζ(s1, ..., sr) = I(0 . . . 01︸︷︷︸s1

0 . . . 01︸︷︷︸s2

. . . 0 . . . 01︸︷︷︸sr

).

In particular, the MZV are Kontsevich-Zagier periods.

14.4. SHUFFLE AND STUFFLE RELATIONS FOR MZV 275

Proof. We will define more generally

I(0; εn . . . ε1; z) :=

∫0≤t1≤···≤tn≤z

ωεn(tn)ωεn−1(tn−1) · · ·ωε1(t1)

for 0 ≤ z ≤ 1. Then we show that

I(0; 0 . . . 01︸︷︷︸s1

0 . . . 01︸︷︷︸s2

. . . 0 . . . 01︸︷︷︸sr

; z) =∑

n1>n2>...>nr≥1

zn1

ns11 · · ·nsrr.

Convergence is always ok for z < 1, but at the end we will have it for z = 1be Abel’s theorem. We proceed by induction on n =

∑ri=1 si. We start with

n = 1:

I(0; 1; z) =

∫ z

0

ω1(t) =

∫ z

0

∑n≥0

tndt =∑n≥0

zn+1

n+ 1=∑n≥1

zn

n.

The induction step has two cases:

I(0; 0 0 . . . 01︸︷︷︸s1

0 . . . 01︸︷︷︸s2

. . . 0 . . . 01︸︷︷︸sr

; z) =

∫ z

0

dtntnI(0; 0 . . . 01︸︷︷︸

s1

0 . . . 01︸︷︷︸s2

. . . 0 . . . 01︸︷︷︸sr

; tn)

=

∫ z

0

dtntn

∑n1>n2>...>nr≥1

tn1n

ns11 · · ·nsrr

=∑

n1>n2>...>nr≥1

zn1

ns1+11 · · ·nsrr

.

I(0; 1 0 . . . 01︸︷︷︸s1

0 . . . 01︸︷︷︸s2

. . . 0 . . . 01︸︷︷︸sr

; z) =

∫ z

0

dtn1− tn

I(0; 0 . . . 01︸︷︷︸s1

0 . . . 01︸︷︷︸s2

. . . 0 . . . 01︸︷︷︸sr

; tn)

=

∫ z

0

dtn

∞∑m=0

tmn∑

n1>n2>...>nr≥1

tn1n

ns11 · · ·nsrr

=

∞∑m=0

∑n1>n2>...>nr≥1

∫ z

0

dtntn1+mn

ns11 · · ·nsrr

=∑

n0>n1>n2>...>nr≥1

zn0

ns11 · · ·nsrr.

In the latter step we strictly use z < 1 to have convergence. It does not occur atthe end of the induction, since the string starts with a 0. Convergence is provenby Abel’s theorem at the end.

14.4 Shuffle and Stuffle relations for MZV

In this section, we present a slightly more abstract viewpoint on multiple zetavalues and their relations by looking only at the strings representing a MZVintegral. It turns out that there are two types of multiplications on those strings,called the shuffle and stuffle products, which induce the usual multiplication on


the integrals, but which have a different definition. Comparing both leads toall kind of relations between multiple zeta values. The reader may also consult[IKZ, Hof, HO, He] for more information.

A MZV can be represented via a tuple (s1, ..., sr) of integers or a string

s = 0 . . . 01︸︷︷︸s1

0 . . . 01︸︷︷︸s2

. . . 0 . . . 01︸︷︷︸sr

of 0’s and 1’s. There is a one-to-one correspondence between strings with a 0on the left and a 1 on the right and all tuples (s1, ..., sr) with all si ≥ 1 ands1 ≥ 2. For any tuple s = (s1, ..., sr), we denote the associated string by s. Wewill formalize the algebras arising from this set-up.

Definition 14.4.1 (Hoffman Algebra). Let

h := Q〈x, y〉 = Q⊕Qx⊕Qy ⊕Qxy ⊕Qyx⊕ · · ·

be the free non-commutative graded algebra in two variables x, y (both of degree1). There are subalgebras

h1 := Q⊕ hy, h0 := Q⊕ xhy.

The generator in degree 0 is denoted by I.

We will now identify x and y with 0 and 1, if it is convenient. For example anygenerator, i.e., a noncommutative word in x and y of length n can be viewed as astring εn · · · ε1 in the letters 0 and 1. With this identification, there is obviouslyan evaluation map such that

ζ : h −→ R, εn · · · ε1 7→ I(εn, ..., ε1)

holds on the generators of h. In addition, if s is the string

s = εn · · · ε1 = 0 . . . 01︸︷︷︸s1

0 . . . 01︸︷︷︸s2

. . . 0 . . . 01︸︷︷︸sr

,

then we have ζ(s1, ..., sn) = ζ(s) by Theorem 14.3.2.

We will now define two different multiplications

X, ∗ : h× h −→ h,

called shuffle and stuffle, such that ζ becomes a ring homomorphism in bothcases.

Definition 14.4.2. Define the shuffle permutations for r + s = n as

Σr,s := σ ∈ Σn | σ(1) < σ(2) < · · · < σ(r), σ(r+1) < σ(r+2) < · · · < σ(r+s).

Define the action of σ ∈ Σr,s on the set 1, 2, ..., n as

σ(x1...xn) := xσ−1(1)...xσ−1(n).


The shuffle product is then defined as

x1...xrXxr+1...xn :=∑

σ∈Σr,s

σ(x1...xn).

Theorem 14.4.3. The shuffle product X defines an associative, bilinear op-eration with unit I and hence an algebra structure on h such that ζ is a ringhomomorphism. It satisfies the recursive formula

uXv = a(u′Xv) + b(uXv′),

if u = au′ and v = bv′ as strings.

Proof. We only give a proof for the product formula ζ(aXb) = ζ(a)ζ(b); therest is straightforward. Assume a = (a1, ..., ar) is of weight m and b = (b1, ..., bs)is of weight n. Then, by Fubini, the product ζ(a)ζ(b) is an integral over theproduct domain

∆ = 0 ≤ t1 ≤ · · · ≤ tm ≤ 1 × 0 ≤ tm+1 ≤ · · · ≤ tm+n ≤ 1.

Ignoring subsets of measure zero,

∆ =∐σ

∆σ

indexed by all shuffles σ ∈ Σr,s, and where

∆σ = (t1, ..., tm+s | 0 ≤ tσ−1(1) ≤ · · · ≤ tσ−1(n) ≤ 1.

The proof follows then from the additivity of the integral.

This induces binary relations as in the following examples.

Example 14.4.4. One has

(01)X(01) = 2(0101) + 4(0011)

and hence we haveζ(2)2 = 2ζ(2, 2) + 4ζ(3, 1).

In a similar way,

(01)X(001) = (010011) + 3(001011) + 9(000111) + (001101),

which implies that

ζ(2)ζ(3, 1) = ζ(2, 3, 1) + 3ζ(3, 2, 1) + 9ζ(4, 1, 1) + ζ(3, 1, 2),

and(01)X(011) = 3(01011) + 6(00111) + (01101)

implies that

ζ(2)ζ(2, 1) = 3ζ(2, 2, 1) + 6ζ(3, 1, 1) + ζ(2, 1, 2).


Definition 14.4.5. The stuffle product

∗ : h× h −→ h

is defined on tuples a = (a1, ..., ar) and b = (b1, ..., bs) as

a ∗ b : = (a1, ..., ar, b1, ..., bs) + (a1, ..., ar + b1, ..., bs)

+ (a1, ..., ar−1, b1, ar, b2, ..., bs) + (a1, ..., ar−1 + b1, ar, b2, ..., bs) + · · ·

The definition is made such that one has the formula ζ(a)ζ(b) = ζ(a ∗ b) in theformula defining multiple zeta values.

Theorem 14.4.6. The stuffle product ∗ defines an associative, bilinear multi-plication on h inducing an algebra (h, ∗) with unit I. One has ζ(a)ζ(b) = ζ(a∗b)on tuples a and b. Furthermore, there is a recursion formula

u ∗ v = (a, u′ ∗ v) + (b, u ∗ v′) + (a, b, u′ ∗ v′)

for tuples u = (a, u′) and v = (b, v′) with first entry a and b.

Proof. Again, we only give a proof for the product formula ζ(a)ζ(b) = ζ(a ∗ b).Assume a = (a1, ..., ar) is of weight m and b = (ar+1, ..., ar+s) is of weight n.The claim follows from a decomposition of the summation range:

ζ(a1, ..., ar)ζ(ar+1, ..., ar+s)

=∑

n1>n2>...>nr≥1

n−a11 · · ·n−arr ·

∑nr+1>nr+2>...>nr+s≥1

n−ar+1

r+1 · · ·n−ar+sr+s =

=∑

n1>n2>...>nr>nr+1>nr+2>...>nr+s≥1

n−a11 · · ·n−arr n

−ar+1

r+1 · · ·n−ar+sr+s

+∑

n1>n2>...>nr=nr+1>nr+2>...>nr+s≥1

n−a11 · · ·n−(ar+ar+1)

r · · ·n−ar+sr+s

+ etc.

where all terms in the stuffle set occur once.

This induces again binary relations as in the following examples.

Example 14.4.7.

ζ(2)ζ(3, 1) = ζ(2, 3, 1) + ζ(5, 1) + ζ(3, 2, 1) + ζ(3, 3) + ζ(3, 1, 2)

ζ(2)2 = 2ζ(2, 2) + ζ(4).

More generally,

ζ(a)ζ(b) = ζ(a, b) + ζ(a+ b) + ζ(b, a), for a, b ≥ 2..


Since we have ζ(aXb) = ζ(a∗ b) we can define the unary double-shuffle relationas

ζ(aXb− a ∗ b) = 0.

Example 14.4.8. We have ζ(2)2 = 2ζ(2, 2)+4ζ(3, 1) using shuffle and ζ(2)2 =2ζ(2, 2) + ζ(4) using the stuffle. Therefore one has

4ζ(3, 1) = ζ(4).

In the literature [Hof, HO, IKZ, He] more relations were found, e.g., a modifiedversion of this relation, called the regularized double-shuffle relation:

ζ

∑b∈(1)∗a

b−∑

c∈(1)Xa

c

= 0.

Example 14.4.9. Let a = (2) = (01). Then (1)X(01) = (101) + 2(011) and(1) ∗ (2) = (1, 2) + (3) + (2, 1). Therefore, the corresponding relation is

ζ(1, 2) + 2ζ(2, 1) = ζ(1, 2) + ζ(3) + ζ(2, 1), hence

ζ(2, 1) = ζ(3).

Like in this example, all non-convergent contributions cancel in the relation. Itis conjectured that the regularized double-shuffle relation generates all relationsamong MZV. There are more relations: the sum theorem (mentioned above),the duality theorem, the derivation theorem and Ohno’s theorem, which impliesthe first three [HO, He].

We will finish this subsection with some formulas mentioned by Brown [B1],mainly due to Broadhurst and Zagier:

ζ(3, 1, . . . , 3, 1︸︷︷︸2n

) =1

2n+ 1ζ(2, 2, . . . , 2︸︷︷︸

2n

) =2π4n

(4n+ 2)!.

ζ(2, ..., 2︸︷︷︸b

, 3, 2, ..., 2︸︷︷︸a

) =∑

m+r=a+b+1

cm,r,a,bπ2m

(4m+ 1)!ζ(2r + 1),

where cm,r,a,b = 2(−1)r((

2r2a+2

)−(1− 2−2r

) (2r

2b+1

))∈ Q (m ≥ 0, r ≥ 1).

In the next section, we relate multiple zeta values to Nori motives and also tomixed Tate motives. This give a more conceptual embedding of such periods inthe sense of Chapter 10, see in particular Section 10.5.


14.5 Multiple zeta values and moduli space ofmarked curves

In this short section, we indicate how one can relate multiple zeta values to Norimotives and to mixed Tate motives.

Multiple zeta values can also be seen as periods of certain cohomology groups ofmoduli spaces in such a way that they appear naturally as Nori motives. Recallthat the moduli space M0,n of smooth rational curves with n marked pointscan be compactified to the space M0,n of stable curves with n markings [K2].Manin and Goncharov [GM] have observed the following.

Theorem 14.5.1. For each convergent multiple zeta value p = ζ(s1, ...., sr) ofweight n = s1 + ...+ sr, one can construct divisors A,B in M0,n+3 such that pis a period of the cohomology group Hn(M0,n+3 \A,B \ (A ∩B)).

The group Hn(M0,n+3 \A,B \ (A∩B)) defines of course immediately a motivein Nori’s sense.

Example 14.5.2. The fundamental example is ζ(2), which we already de-scribed in section 14.1. Here M0,5 is a compactification of

M0,5 = (P \ 0, 1,∞)2 \ diagonal,

since M0,5 is the blow up (0, 0), (1, 1) and (∞,∞) in P1×P1. This realizes ζ(2)as the integral

ζ(2) =

∫0≤t1≤t2≤1

dt11− t2

dt2t2

.

We leave it to the reader to make the divisors A and B explicit.

This viewpoint was very much refined in Brown’s thesis [B3]. Recent relatedresearch for higher polylogarithms and elliptic polylogarithms can be found in[B4].

Levine [L2] has defined an abelian category as a full subcategory of the tri-angulated category of geometrical motives, see Chapter 10 for the notion ofgeometric motives. It is a full subcategory generated by the Tate objects Q(n).There is also a variant, called mixed Tate motives over Z, see [Te, DG, B1]. TheTheorem above implies:

Theorem 14.5.3 (Brown). Multiple zeta values together with (2πi)n are pre-cisely all the periods of all mixed Tate motives over Z.

Proof. This is a result of Brown, see [B1, D3].

14.6. MULTIPLE POLYLOGARITHMS 281

14.6 Multiple Polylogarithms

In this section, we study a variation of cohomology groups in a 2-parameterfamily of varieties over Q, the so-called double logarithm variation, for whichmultiple polylogarithms appear as coefficients. This viewpoint gives more exam-ples of Kontsevich-Zagier periods occuring as cohomological periods of canonicalcohomology groups at particular values of the parameters. The degeneration ofthe parameters specializes such periods to simpler ones.

First define the hyperlogarithm as the iterated integral

In(a1, . . . , an) :=

∫0≤t1≤···≤tn≤1

dt1t1 − a1

∧ · · · ∧ dtntn − an

with a1, . . . , an ∈ C (cf. [Z1, p. 168]). Note that, the order of terms here isdifferent from the previous order, also in the infinite sum below.

These integrals specialize to the multiple polylogarithm (cf. [loc. cit.])

Lim1,...,mn

(a2

a1, · · · , an

an−1,

1

an

):= (−1)n I∑mn(a1, 0, . . . , 0︸︷︷︸

m1−1

, . . . , an, 0, . . . , 0︸︷︷︸mn−1

),

which is convergent if 1 < |a1| < · · · < |an| (cf. [G3, 2.3, p. 9]). Alternatively,we can describe the multiple polylogarithm as a power series (cf. [G3, Thm. 2.2,p. 9])

Lim1,...,mn(x1, . . . , xn) =∑

0<k1<···<kn

xk11 · · ·xknn

km11 · · · kmnn

for |xi| < 1. (14.1)

Of special interest to us will be the dilogarithm Li2(x) =∑k>0

xk

k2 and the

double logarithm Li1,1(x, y) =∑

0<k<lxkyl

kl .

Remark 14.6.1. At first, the functions Lim1,...,mn(x1, . . . , xn) only make sensefor |xi| < 1, but they can be analytically continued to multivalued meromorphicfunctions on Cn (cf. [Z1, p. 2]), for example Li1(x) = − log(1 − x). One has

Li2(1) = π2

6 .

14.6.1 The Configuration

Let us consider the configuration

Y := A2 with coordinates x and y,

Z := x = a ∪ y = b with a 6= 0, 1 and b 6= 0, 1

X := Y \ ZD := (x = 0 ∪ y = 1 ∪ x = y) \ Z,


see Figure 14.3.

We denote the irreducible components of the divisor D as follows:

D1 := x = 0 \ (0, b),D2 := y = 1 \ (a, 1), and

D3 := x = y \ (a, a), (b, b).

By projecting from Y onto the y- or x-axis, we get isomorphisms for the asso-ciated complex analytic spaces

Dan1∼= C \ b, Dan

2∼= C \ a, and Dan

3∼= C \ a, b.

Figure 14.3: The algebraic pair (X,D)

14.6.2 Singular Homology

We can easily give generators for the second singular homology of the pair(X,D), see Figure 14.4.

• Let α : [0, 1] → C be a smooth path, which does not meet a or b. Wedefine a “triangle”

4 := (α(s), α(t)

)| 0 ≤ s ≤ t ≤ 1.

• Consider the closed curve in C

Cb :=

a

b+ εe2πis| s ∈ [0, 1]

,

which divides C into two regions: an inner one containing ab and an outer

one. We can choose ε > 0 small enough such that Cb separates ab from 0

to 1, i.e., such that 0 and 1 are contained in the outer region. This allows


us to find a smooth path β : [0, 1] → C from 0 to 1 not meeting Cb. Wedefine a “slanted tube”

Sb :=(β(t) · (b+ εe2πis), b+ εe2πis

)| s, t ∈ [0, 1]

which winds around y = b and whose boundary components are sup-ported on D1 (corresponding to t = 0) and D3 (corresponding to t = 1).The special choice of β guarantees Sb ∩ Z(C) = ∅.

• Similarly, we choose ε > 0 such that the closed curve

Ca :=

b− 1

a− 1− εe2πis| s ∈ [0, 1]

separates b−1

a−1 form 0 and 1. Let γ : [0, 1] → C be a smooth path from 0to 1 which does not meet Ca. We have a “slanted tube”

Sa :=(a+ εe2πis, 1 + γ(t) · (a+ εe2πis − 1)

)| s, t ∈ [0, 1]

winding around x = a with boundary supported on D2 and D3.

• Finally, we have a torus

T := (a+ εe2πis, b+ εe2πit) | s, t ∈ [0, 1].

The 2-form ds∧dt defines an orientation on the unit square [0, 1]2 = (s, t) | s, t ∈[0, 1]. Hence the manifolds with boundary 4, Sb, Sa, T inherit an orientation,and since they can be triangulated, they give rise to smooth singular chains.By abuse of notation we will also write 4, Sb, Sa, T for these smooth singularchains. The homology classes of 4, Sb, Sa and T will be denoted by γ0, γ1, γ2

and γ3, respectively.

Figure 14.4: Generators of Hsing2 (X,D;Q)


An inspection of the long exact sequence in singular homology will reveal thatγ0, . . . , γ3 form a system generators (see the following proof)

Hsing2 (D,Q) −−−−→ Hsing

2 (X,Q) −−−−→ Hsing2 (X,D,Q) −−−−→

Hsing1 (D,Q)

i1−−−−→ Hsing1 (X,Q) .

Proposition 14.6.2. With notation as above, we have for the second singularhomology of the pair (X,D)

Hsing2 (X,D;Q) = Q γ0 ⊕Q γ1 ⊕Q γ2 ⊕Q γ3.

Proof. For c := a and c := b, the inclusion of the circle c + εe2πis | s ∈ [0, 1]into C \ c is a homotopy equivalence, hence the product map T → X(C) isalso a homotopy equivalence. This shows

Hsing2 (X,Q) = QT,

while Hsing1 (X,Q) has rank two with generators

• one loop winding counterclockwise around x = a once, but not aroundy = b, thus being homologous to both ∂Sa ∩D2(C) and −∂Sa ∩D3(C),and

• another loop winding counterclockwise around y = b once, but notaround x = a, thus being homologous to ∂Sb∩D1(C) and −∂Sb∩D3(C).

In order to compute the Betti-numbers bi of D, we use the spectral sequencefor the closed covering Di

Ep,q2 :=

· · · 0 0 0 0 · · ·· · · 0

⊕3i=1H

1dR(Di,C) 0 0 · · ·

· · · 0 Kerδ Cokerδ 0 · · ·· · · 0 0 0 0 · · ·

⇒ En∞ := HndR(D,C),

where

δ :

3⊕i=1

H0dR(Di,C) −→

⊕i<j

H0dR(Dij ,C).

Note that this spectral sequence degenerates. Since D is connected, we haveb0 = 1, i.e.,

1 = b0 = rankCE0∞ = rankCE

0,02 = rankCKerδ.

Hence

rankCCokerδ = rankCcodomain δ − rankCdomain δ + rankCKerδ

= (1 + 1 + 1)− (1 + 1 + 1) + 1 = 1,


and so

b1 = rankCE1∞ = rankCE

1,02 + rankCE

0,12

=

3∑i=1

rankCH1dR(Di,C) + rankCCokerδ

= rankCH1(C \ b,C) + rankCH

1(C \ a,C) + rankCH1(C \ a, b,C) + 1

= (1 + 1 + 2) + 1 = 5.

We can easily specify generators of Hsing1 (D,Q) as follows

Hsing1 (D,Q) = Q·(∂Sb∩D1)⊕Q·(∂Sa∩D2)⊕Q·(∂Sb∩D3)⊕Q·(∂Sa∩D3)⊕Q·∂4.

Clearly b2 = rankCHsing2 (D,Q) = 0. Now we can compute Keri1 and obtain

Keri1 = Q·∂4⊕Q·(∂Sb∩D1(C)+∂Sb∩D3(C))⊕Q·(∂Sa∩D2(C)+∂Sa∩D3(C)).

This shows finally

rankQHsing2 (X,D;Q) = rankQH

sing2 (X,Q) + rankQKeri1 = 1 + 3 = 4.

From these explicit calculations we also derive the linear independence of γ0 =[4], γ1 = [Sb], γ2 = [Sa], γ3 = [T ] and Proposition 14.6.2 is proved.

14.6.3 Smooth Singular Homology

Recall the definition of smooth singular cohomology (cf. Theorem 2.2.5). Withthe various sign conventions made so far, the boundary map δ : C∞2 (X,D;Q)→C∞1 (X,D;Q) is given by

δ : C∞2 (X,Q)⊕3⊕i=1

C∞1 (Di,Q)⊕⊕i<j

C∞0 (Dij ,Q)→ C∞1 (X,Q)⊕3⊕i=1

(Di,Q)

(σ∅, σ1

1, σ2

2, σ3

3, σ12

12, σ13

13, σ23

23) 7→

(∂σ + σ1 + σ2 + σ3∅

,−∂σ1 + σ12 + σ131

,−∂σ2 − σ12 + σ232

,−∂σ3 − σ13 − σ233

).

Thus the following elements of C∞2 (X,D;Q) are cycles

• Γ0 := (4∅,−∂4∩D1(C)

1,−∂4∩D2(C)

2,−∂4∩D3(C)

3, D12(C)

12,−D13(C)

13, D23(C)

23),

• Γ1 := (Sb∅,−∂Sb ∩D1(C)

1, 0

2,−∂Sb ∩D3(C)

3, 012, 013, 023

),

• Γ2 := (Sa∅, 0

1,−∂Sa ∩D2(C)

2, 0

3,−∂Sa ∩D3(C)

12, 013, 023

) and

• Γ3 := (T∅, 0

1, 0

2, 0

3, 012, 013, 023

).

Under the isomorphism H∞2 (X,D;Q)∼−→ Hsing

2 (X,D;Q) the classes of thesecycles [Γ0], [Γ1], [Γ2], [Γ3] are mapped to γ0, γ1, γ2, γ3, respectively.


14.6.4 Algebraic de Rham cohomology and period matrixof (X,D)

Recall the definition of the complex Ω•X,D. We consider

Γ(X, Ω2X,D) = Γ(X,Ω2

X)⊕3⊕i=1

Γ(Di,Ω1Di)⊕

⊕i<j

Γ(Dij ,ODij )

together with the following cycles of Γ(X, Ω2X,D)

• ω0 := ( dx∧dy(x−a)(y−b)

∅

, 01, 0

2, 0

3, 012, 013, 023

),

• ω1 := (0∅, −dyy−b

1

, 02, 0

3, 012, 013, 023

),

• ω2 := (0∅, 0

1, −dxx−a

2

, 03, 012, 013, 023

), and

• ω3 := (0∅, 0

1, 0

2, 0

3, 012, 013, 123

).

By computing the (transposed) period matrix Pij := 〈Γj , ωi〉 and checking itsnon-degeneracy, we will show that ω0, . . ., ω3 span H2

dR(X,D).

Proposition 14.6.3. Let X and D be as above. Then the second algebraic deRham cohomology group H2

dR(X,D) of the pair (X,D) is generated by the cyclesω0, . . . , ω3 considered above.

Proof. Easy calculations give us the (transposed) period matrix P :

Γ0 Γ1 Γ2 Γ3

ω0 1 0 0 0ω1 Li1( 1

b ) 2πi 0 0ω2 Li1( 1

a ) 0 2πi 0

ω3 ? 2πiLi1( ba ) 2πi log(a−b1−b

)(2πi)2.

For example,

• P1,1 = 〈Γ1, ω1〉 =∫−∂Sb∩D1(C)

−dyy−b

=∫|y−b|=ε

dyy−b

= 2πi,

• P3,3 = 〈Γ3, ω3〉 =∫T

dxx−a ∧

dyy−b

=(∫|x−a|=ε

dxx−a

)·(∫|y−b|=ε

dyy−b

)by Fubini

= (2πi)2,


• P1,0 = 〈Γ0, ω1〉 =∫−∂4∩D1(C)

−dyy−b

=∫ 1

0−α(t)α(t)−b

= −[log(α(t))− b]10= − log

(1−b−b

)= − log

(1− 1

b

)= Li1

(1b

), and

• P3,1 = 〈Γ1, ω3〉 =∫Sb

dxx−a ∧

dyy−b

=∫

[0,1]2d(β(t)·(b+εe2πis))β(t)·(b+εe2πis)−a ∧

d(b+εe2πis)εe2πis

=∫

[0,1]2b+εe2πis

β(t)·(b+εe2πis)−adβ(t) ∧ 2πids

= −∫ 1

0

[a log(β(t)·(b+εe2πis)−a)−2πiβ(t)bs

β(t)·(−β(t)b+a)

]1

0

dβ(t)

= −2πi∫ 1

0dβ(t)β(t)− ab

= −2πi[log(β(t)− a

b

)]10

= −2πi log(

1− ab− ab

)= −2πi log

(1− a

b

)= 2πiLi1

(ba

).

Obviously the period matrix P is non-degenerate and so Proposition 14.6.3 isproved.

What about the entry P3,0?

Proposition 14.6.4. P3,0 = Li1,1(ba ,

1b

).

For the proof we need to show that 〈Γ0, ω3〉 = Li1,1(ba ,

1b

), where Li1,1(x, y)

is an analytic continuation of the double logarithm defined for |x|, |y| < 1 inSubsection 14.6.

Lemma 14.6.5. The integrals

Iα2

(1

xy,

1

y

)=

∫0≤s≤t≤1

dα(s)

α(s)− 1xy

∧ dα(t)

α(t)− 1y

with α : [0, 1] → C a smooth path from 0 to 1, and 1xy ,

1b ∈ C \ Imα, defined

above on page 288, provide a genuine analytic continuation of Li1,1(x, y) to amultivalued function which is defined on (x, y) ∈ C2 |x, y 6= 0, xy 6= 1, y 6= 1.

Proof. We describe this analytic continuation in detail. Our approach is similarto the one taken in [G3, 2.3, p. 9], but differs from that in [Z2a, p. 7].


Let Ban := (C\0, 1)2 be the parameter space and choose a point (a, b) ∈ Ban.For ε > 0 we denote by Dε(a, b) the polycylinder

Dε(a, b) := (a, b) ∈ Ban | |a′ − a| < ε, |b′ − b| < ε.

If α : [0, 1] → C is a smooth path from 0 to 1 passing through neither a nor b,then there exists an ε > 0 such that Imα does not meet any of the discs

D2ε(a) := a′ ∈ C | |a′ − a| < 2ε, and

D2ε(b) := b ′ ∈ C | |b ′ − b| < 2ε.

Hence the power series (14.2) below

1

α(s)− a′1

α(t)− b ′=

1

α(s)− a1

1− a′−aα(s)−a

1

α(t)− b1

1− b ′−bα(t)−b

=

∞∑k,l=0

1

(α(s)− a)k+1(α(t)− b)l+1︸︷︷︸ck.l

(a′ − a)k(b ′ − b)l (14.2)

has coefficients ck,l satisfying

|ck,l| <(

1

2ε

)k+l+2

.

In particular, (14.2) converges uniformly for (a′, b ′) ∈ Dε(a, b) and we see thatthe integral

Iα2 (a′, b ′) :=

∫0≤s≤t≤1

dα(s)

α(s)− a′∧ dα(t)

α(t)− b ′

=∑k,l=0

(∫0≤s≤t≤1

dα(s)

(α(s)− a)k+1∧ dα(t)

(α(t)− b)l+1

)(a′ − a)k(b ′ − b)l

defines an analytic function of Dε(a, b). In fact, by the same argument we getan analytic function Iα2 on all of (C \ Imα)2.

Now let αr : [0, 1]→ C\(D2ε(a) ∪D2ε(b)) with r ∈ [0, 1] be a smooth homotopyof paths from 0 to 1, i.e. αr(0) = 0 and αr(1) = 1 for all r ∈ [0, 1]. We show

Iα02 (a′, b ′) = Iα1

2 (a′, b ′) for all (a′, b ′) ∈ Dε(a, b).

Define a subset Γ ⊂ C2

Γ := (αr(s), αr(t)) | 0 ≤ s ≤ t ≤ 1, r ∈ [0, 1].

The boundary of Γ is built out of five components (each being a manifold withboundary)

• Γs=0 := (0, αr(t)) | r, t ∈ [0, 1],


• Γs=t := (αr(s), αr(s)) | r, s ∈ [0, 1],

• Γt=1 := (αr(s), 1) | r, s ∈ [0, 1],

• Γr=0 := (α0(s), α0(t) | 0 ≤ s ≤ t ≤ 1,

• Γr=1 := (α1(s), α1(t) | 0 ≤ s ≤ t ≤ 1.

Let (a′, b ′) ∈ Dε(a, b). Since the restriction of dxx−a′ ∧

dyy−b ′ to Γs=0, Γs=t and

Γt=1 is zero, we get by Stokes’ theorem

0 =

∫Γ

0 =

∫Γ

ddx

x− a′∧ dy

y − b ′

=

∫∂Γ

dx

x− a′∧ dy

y − b ′

=

∫Γr=1−Γr=0

dx

x− a′dy

y − b ′

= Iα12 (a′, b ′)− Iα0

2 (a′, b ′).

For each pair of smooth paths α0, α1 : [0, 1] → C from 0 to 1, we can find ahomotopy αr relative to 0, 1 between both paths. Since Imαr is compact, wealso find a point (a, b) ∈ Ban = (C \ 0, 1)2 and an ε > 0 such that Imαr doesnot meet D2ε(a, b) or D2ε(a, b). Then Iα0

2 and Iα12 must agree on Dε(a, b). By

the identity principle for analytic functions of several complex variables [Gun,A, 3, p. 5], the functions Iα2 (a′, b ′), each defined on (C \ Imα)2, patch togetherto give a multivalued analytic function on Ban = (C \ 0, 1)2.

Now assume 1 < |b| < |a|, then we can take α = id : [0, 1] → C, s 7→ s, andobtain

Iid2 (a, b) = I2(a, b) = Li1,1

(b

a,

1

y

),

where Li1,1(x, y) is the double logarithm defined for |x|, |y| < 1 in Subsection14.6. Thus we have proved the lemma.

Definition 14.6.6 (Double logarithm). We call the analytic continuation fromLemma 14.6.5 the double logarithm as well and continue to use the notationLi1,1(x, y).

The period matrix P is thus given by:

Γ0 Γ1 Γ2 Γ3

ω0 1 0 0 0ω1 Li1( 1

b ) 2πi 0 0ω2 Li1( 1

a ) 0 2πi 0

ω3 Li1,1(ba ,

1b

)2πiLi1( ba ) 2πi log

(a−b1−b

)(2πi)2.


14.6.5 Varying parameters a and b

The homology groupHsing2 (X,D;Q) of the pair (X,D) carries a Q-MHS (W•, F

•).The weight filtration is given in terms of the γj:

WpHsing2 (X,D;Q) =

0 for p ≤ −5

Qγ3 for p = −4,−3

Qγ1 ⊕Qγ2 ⊕Qγ3 for p = −2,−1

Qγ0 ⊕Qγ1 ⊕Qγ2 ⊕Qγ3 for p ≥ 0,

The Hodge filtration is given in terms of the ω∗i :

F pHsing2 (X,D;Q) =

Cω∗0 ⊕ Cω∗1 ⊕ Cω∗2 ⊕ Cω∗3 for p ≤ −2

Cω∗0 ⊕ Cω∗1 ⊕ Cω∗2 for p = −1

Cω∗0 for p = 0

0 for p ≥ 1.

This Q-MHS resembles very much the Q-MHS considered in [G1, 2.2, p. 620]and [Z2a, 3.2, p. 6]. Nevertheless a few differences are note-worthy:

• Goncharov defines the weight filtration slightly different, for example hislowest weight is −6.

• The entry P3,2 = 2πi log(a−b1−b

)of the period matrix P differs by (2πi)2,

or put differently, the basis γ0, γ1, γ2 − γ3, γ3 is used.

Up to now, the parameters a and b of the configuration (X,D) have been fixed.By varying a and b, we obtain a family of configurations: Equip A2

C with coor-dinates a and b and let

B := A2C \ (a = 0 ∪ a = 1 ∪ b = 0 ∪ b = 1)

be the parameter space. Take another copy of A2C with coordinates x and y and

define total spaces

X := (B × A2C

(a,b,x,y)

) \ (x = a ∪ y = b) , and

D := “B ×D” = X ∩ (x = 0 ∪ y = 1 ∪ x = y) .

We now have a projection

D → X (a, b, x, y)

yπ yB (a, b)

,

whose fiber over a closed point (a, b) ∈ B is precisely the configuration (X,D)for the parameter choice a, b. π is a flat morphism. The assignment

(a, b) 7→ (VQ,W•, F•),


where

VQ := spanQs0, . . . , s3,

VC := C4 with standard basis e0, . . . , e3,

s0 :=

1

Li(

1b

)Li1(

1a

)Li1,1

(ba ,

1b

) , s1 :=

0

2πi0

2πiLi1(ba

) , s2 :=

00

2πi

2πi log(a−b1−b

) , s3 :=

000

(2πi)2

,

WpVQ =

0 for p ≤ −5

Qs3 for p = −4,−3

Qs1 ⊕Qs2 ⊕Qs3 for p = −2,−1

VQ for p ≥ 0, and

F pVC =

VC for p ≤ −2

Ce0 ⊕ Ce1 ⊕ Ce2 for p = −1

Ce0 for p = 0

0 for p ≥ 1

defines a good unipotent variation of Q-MHS on Ban. Note that the Hodgefiltration F • does not depend on (a, b) ∈ Ban.

One of the main characteristics of good unipotent variations of Q-MHS is thatthey can be extended to a compactification of the base space (if the complementis a divisor with normal crossings).

The algorithm for computing these extensions, so called limit mixed Q-Hodgestructures, can be found for example in [H, 7, p. 24f] and [Z2b, 4, p. 12].

In a first step, we extend the variation to the divisor a = 1 minus the point(1, 0) and then in a second step we extend it to the point (1, 0). In particular,we assume that a branch has been picked for each entry Pij of P . We will follow[Z2b, 4.1, p. 14f] very closely.

First step: Let σ be the loop winding counterclockwise around a = 1 once,but not around a = 0, b = 0 or b = 1. If we analytically continue anentry Pij of P along σ we possibly get a second branch of the same multivaluedfunction. In fact, the matrix resulting from analytic continuation of every entryalong σ will be of the form

P · Ta=1,

where

Ta=1 =

1 0 0 00 1 0 0−1 0 1 00 0 0 1


is the monodromy matrix corresponding to σ. The local monodromy logarithmis defined as

Na=1 =log Ta=1

2πi=

1

2πi

∞∑n=1

−1

n

((1

11

1

)− Ta=1

)n

=

0 0 0 00 0 0 0−12πi 0 0 00 0 0 0

.

We want to extend our Q-MHS along the tangent vector ∂∂a , i.e. we introduce

a local coordinate t := a− 1 and compute the limit period matrix

Pa=1 := limt→0

P · e− log(t)·Na=1

= limt→0

1 0 0 0

Li1(

1b

)2πi 0 0

Li1

(1

1+t

)0 2πi 0

Li1,1

(b

1+t ,1b

)2πiLi1

(b

1+t

)2πi log

(1−b+t

1−b

)(2πi)2

·

1 0 0 00 1 0 0

log(t)2πi 0 1 00 0 0 1

= limt→0

1 0 0 0

Li1(

1b

)2πi 0 0

Li1

(1

1+t

)+ log(t) 0 2πi 0

Li1,1

(b

1+t ,1b

)+ log

(1−b+t

1−b

)· log(t) 2πiLi1

(b

1+t

)2πi log

(1−b+t

1−b

)(2πi)2

(∗)=

1 0 0 0

Li1(

1b

)2πi 0 0

0 0 2πi 0

−Li2

(1

1−b

)2πiLi1(b) 0 (2πi)2

.

Here we used at (∗)

• Pa=12,0 = limt→0 Li1

(1

1+t

)+ log(t)

= limt→0− log(

1− 11+t

)+ log(t)

= limt→0− log(t) + log(1 + t) + log(t)

= 0, and

• Pa=13,0 = limt→0 Li1,1

(b

1+t ,1b

)+ log

(1−b+t

1−b

)· log(t)

= Li1,1(b, 1b

)by L’Hospital

= −Li2

(1

1− b

).


The vectors s0, s1, s2, s3 spanning the Q-lattice of the limit Q-MHS on a =1 \ (1, 0) are now given by the columns of the limit period matrix

s0 =

1

Li1(

1b

)0

−Li2

(1

1−b

) , s1 =

0

2πi0

2πiLi1(b)

, s2 =

00

2πi0

, s3 =

000

(2πi)2

.

The weight and Hodge filtration of the limit Q-MHS can be expressed in termsof the sj and the standard basis vectors ei of C4. This gives us a variation ofQ-MHS on the divisor a = 1\(1, 0). This variation is actually (up to signs)an extension of Deligne’s famous dilogarithm variation considered for examplein [Kj, 4.2, p. 38f]. In loc. cit. the geometric origin of this variation is explainedin detail.

Second step: We now extend this variation along the tangent vector −∂∂b to thepoint (1, 0), i.e. we write b = −t with a local coordinate t. Let σ be the loop ina = 1 \ (1, 0) winding counterclockwise around (1, 0) once, but not around(1, 1). Then the monodromy matrix corresponding to σ is given by

T(1,0) =

1 0 0 01 1 0 00 0 1 00 0 0 1

,

hence the local monodromy logarithm is given by

N(1,0) =log T(1,0)

2πi=

0 0 0 01

2πi 0 0 00 0 0 00 0 0 0

.

Thus we get for the limit period matrix

P(1,0) := limt→0

Pa=1 · e− log(t)·N(1,0)

= limt→0

1 0 0 0

Li1(−1t

)2πi 0 0

0 0 2πi 0

−Li2

(1

1+t

)2πiLi1(−t) 0 (2πi)2

·

1 0 0 0− log(t)

2πi 1 0 00 0 1 00 0 0 1

= limt→0

1 0 0 0

Li1(−1t

)− log(t) 2πi 0 0

0 0 2πi 0

−Li2

(1

1+t

)− Li1(−t) · log(t) 0 0 (2πi)2

(∗)=

1 0 0 00 2πi 0 00 0 2πi 0

−ζ(2) 0 0 (2πi)2

.


We remark that in the last matrix we see a decomposition into two (2×2)-blocks,one consisting of a Tate motive, the other involving ζ(2).

Here we used at (∗)

• P(1,0)1,0= limt→0 Li1

(−1t

)− log(t)

= limt→0− log(1 + 1

t

)− log(t)

= limt→0− log(1 + t) + log(t)− log(t)

= 0, and

• P(1,0)3,0= limt→0−Li2

(1

1+t

)− Li1(−t) · log(t)

= limt→0 Li2

(1

1+t

)+ log(1 + t) · log(t)

= −Li2(1) by L’Hospital

= −ζ(2).

As in the previous step, the vectors s0, s1, s2, s3 spanning the Q-lattice of thelimit Q-MHS are given by the columns of the limit period matrix P(1,0) andweight and Hodge filtrations by the formulae in subsection 14.6.5.

So we obtained −ζ(2) as a “period” of a limiting Q-MHS.

Chapter 15

Miscellaneous periods: anoutlook

In this chapter, we collect several other important examples of periods in theliterature for the convenience of the reader.

15.1 Special values of L-functions

The Beilinson conjectures give a formula for the values (more precisely, theleading coefficients) of L-functions of motives at integral points. We sketch theformulation in order to explain that these numbers are periods.

In this section, fix the base field k = Q. Let Γ = Gal(Q/Q) be the absoluteGalois group. For any prime p, let Ip ⊂ Γ be the inertia group. Let Frp ∈ Γ/Ipbe the Frobenius.

Let M be a mixed motive, i.e., an object in the conjectural Q-linear abeliancategory of mixed motives over Q. For any prime l, it has an l-adic realizationMl which is a finite dimensional Ql-vector space with a continuous operation ofthe absolute Galois group GQ = Gal(Q/Q).

Definition 15.1.1. Let M as above, p a prime and l a prime different from p.We put

Pp(M, t)l = det(1− Frpt|MIpl ) ∈ Ql[t] .

It is conjectured that Pp(M, t)l is in Q[t], and independent of l. We denote thispolynomial by Pp(M, t).

Example 15.1.2. Let M = Hi(X) for smooth projective variety over Q withgood reduction at p. Then the conjecture holds by the Weil conjectures proved

295

296 CHAPTER 15. MISCELLANEOUS PERIODS: AN OUTLOOK

by Deligne. In the special case X = Spec(Q), we get

Pp(H(SpecQ), t) = 1− t .

In the special case X = P1, i = 1, we get

Pp(H1(P1), t) = 1− pt .

Remark 15.1.3. There is a sign issue with the operation of Frp depending onthe normalization of Fr ∈ Gal(Fp/Fp) and whether it operates via geometric orarithmetic Frobenius. We refrain from working out all the details.

Definition 15.1.4. Let M be as above. We put

L(M, s) =∏

p prime

1

Pp(M,p−s)

as function in the variable s ∈ C. For n ∈ Z, let

L(M,n)∗

be the leading coefficient of the Laurent expansion of L(M, s) around n.

We conjecture that the infinite product converges for Re(s) big enough and thatthe function has a meromorphic continuation to all of C.

Example 15.1.5. Let M = Hi(X) for X a smooth projective variety overQ. Then convergence follows from the Riemann hypothesis part of the Weilconjectures. (Note that X has good reduction at almost all p. It suffices toconsider these. Then the zeros of Pp(M, t) are known to have absolute value

p−i2 .

Analytic continuation is a very deep conjecture. It holds for all 0-dimensionalX. Indeed, for any number field K, we have

L(H0(SpecK), s) = ζK(s)

where ζK(s) is the Dedekind ζ-function. For M = H1(E) with E an ellipticcurve over Q, we have

L(H1(E), s) = L(E, s) .

Analytic continuation holds, because E is modular.

Example 15.1.6. Let M be as above, Q(−1) = H2(P1) be the Lefschetz mo-tive. We put M(−1) = M ⊗Q(−1). Then

L(M(−1), s) = L(M, s− 1)

by the formula for Pp(Q(−1), t) above.

15.1. SPECIAL VALUES OF L-FUNCTIONS 297

Hence, the Beilinson conjecture about L(M, s) at n ∈ Z can be reduced to theBeilinson conjecture about L(M(−n), s) at 0.

Conjecture 15.1.7 (Beilinson [Be3]). Let M be as above. Then the vanishingorder of L(M, s) at s = 0 is given by

dimH1M,f (SpecQ,M∗(1))− dimH0

M,f (SpecQ,M),

where HM,f is unramified motivic cohomology.

For a conceptional discussion of unramified motivic cohomology and a compar-ison of the different possible definitions, see Scholbach’s discussion in [Sch2].

In particular, we assume that unramified motivic cohomology is finite dimen-sional.

This conjecture is known for example when M = H0(SpecK)(n) with K anumber field, n ∈ Z or when M = H1(E) with E an elliptic curve with Mordell-Weil rank at most 1.

Definition 15.1.8. We call M special if the motivic cohomology groups

H0M,f (SpecQ,M), H1

M,f (SpecQ,M), H0M,f (SpecQ,M∗(1)), H1

M,f (SpecQ,M∗(1))

all vanish.

We are only going to state the Beilinson conjecture for special motives. In thiscase it is also known as Deligne conjecture. This suffices:

Proposition 15.1.9 (Scholl, [Scho]). Let M be a motive as above. Assumeall motivic cohomology groups over Q are finite-dimensional. Then there is aspecial motive M ′ such that

L(M, 0)∗ = L(M ′, 0)

and the Beilinson conjecture for M is equivalent to the Beilinson conjecture forM ′.

Conjecture 15.1.10 (Beilinson [Be3], Deligne [D1]). Let M be a special motive.Let MB be its Betti-realization and MdR its de Rham realization.

1. L(M, 0) is defined and non-zero.

2. The composition

M+B ⊗ C→MB ⊗ C per−−→MdR ⊗ C→MdR ⊗ C/F 0MdR ⊗ C

is an isomorphism. Here M+B denotes the invariants under complex con-

jugation and F 0MdR denotes the 0-step of the Hodge filtration.

3. Up to a rational factor, the value L(M, 0) is given by the determinant ofthe above isomorphism in any choice of rational basis of M+

B and MdR.


Corollary 15.1.11. Assume the Beilinson conjecture holds. Let M be a motive.Then L(M, 0)∗ is a period number.

Proof. By Scholl’s reduction, it suffices to consider the case M special. Thematrix of the morphism in the conjecture is a block in the matrix of

per : MB ⊗ C→MdR ⊗ C .

All its entries are periods. Hence, the same is true for the determinant.

15.2 Feynman periods

Standard procedures in quantum field theory (QFT) lead to loop amplitudesassociated to certain graphs [BEK, MWZ2]. Although the foundations of QFTvia path integrals are mathematically non-rigorous, Feynman and others haveset up the so-called Feynman rules as axioms, leading to a mathematicallyprecise definition of loop integrals (or, amplitudes).

These are defined as follows. Associated to a graph G one defines the integralas

IG =

∏nj=1 Γ(νj)

Γ(ν − `D/2)

∫RD`

∏`r=1 dkriπD/2

n∏j=1

(−q2j +m2

j )−νj .

Here, D is the dimension of space-time (usually, but not always, D = 4), n isthe number of internal edges of G, ` = h1(G) is the loop number, νj are integersassociated to each edge, ν is the sum of all νj , the mj are masses, the qj arecombinations of external momenta and internal loop momenta kr, over whichone has to integrate [MWZ2, Sect. 2]. All occurring squares are scalar productsin D-dimensional Minkowski space. The integrals usually do not converge inD-space, but standard renormalization procedures in physics, e.g. dimensionalregularization, lead to explicit numbers as coefficients of Laurent series. Indimensional regularization, one views the integrals as analytic meromorphicfunctions in the paramter ε ∈ C where D = 4 − 2ε. The coefficients of theresulting Laurent expansion in the variable ε are then the relevant numbers. Bya theorem of Belkale-Brosnan [BB] and Bogner-Weinzierl [BW], such numbersare periods, if all moments and masses in the formulas are rational numbers.

A process called Feynman-Schwinger trick [BEK] transforms the above integralinto a period integral

IG =

∫σ

fω

with

f =

∏nj=1 x

νj−1j Uν−(`+1)D/2

Fν−`D/2, ω =

n∑j=1

(−1)jdx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxn.

15.2. FEYNMAN PERIODS 299

Here, U and F are homogenous graph polynomials of Kirchhoff type [MWZ2,Sect. 2], with only F depending on kinematical invariants, and σ is the standardreal simplex in Pn−1(C). Since σ is a compact subset of Pn−1(C), this is almosta representation of IG as a naive period, and it is indeed one as a Kontsevich-Zagier period, provided the external momenta pi are rational numbers. Thedifferential form fω has poles along σ, but there is a canonical blow-up processto resolve this problem [BEK, MWZ2]. The period which emerges is the periodof the relative cohomology group

Hn(P \ Y,B \ (B ∩ Y )),

where P is a blow-up of projective space in linear coordinate subspaces, Y isthe strict transform of the singularity set of the integrand, and B is the stricttransform of the standard algebraic simplex ∆n−1 ⊂ Pn−1 [MWZ2, Sect. 2].It is thus immediate that IG is a Kontsevich-Zagier period, if it is convergent,and provided that all masses and momenta involved are rational. If IG is notconvergent, then, by a theorem of Belkale-Brosnan [BB] and Bogner-Weinzierl[BW], the same holds under these assumptions for the coefficients of the Laurentexpansion in renormalization.

Example 15.2.1. A very popular graph with a divergent amplitude is the two-loop sunset graph

p

&%'$

The corresponding amplitude in D dimensions is

Γ(3−D)

∫σ

(x1x2 + x2x3 + x3x1)3− 32D(x1dx2 ∧ dx3 − x2dx1 ∧ dx3 + x3dx1 ∧ dx2)

(−x1x2x3p2 + (x1m21 + x2m2

2 + x3m23)(x1x2 + x2x3 + x3x1))3−D ,

where σ is the real 2-simplex in P2.

In D = 4, this integral does not converge. One may, however, compute theintegral in D = 2 and study its dependence on the momentum p as an inho-mogenous differential equation [MWZ1]. There is an obvious family of ellipticcurves involved in the equations of the denominator of the integral, which givesrise to the homogenous Picard-Fuchs equation [MWZ1]. Then, a trick of Tarasovallows to compute the D = 4 situation from that, see [MWZ1]. The extensionof mixed Hodge structures

0→ Z(−1)→ H2(P \ Y,B \B ∩ Y )→ H2(P \ Y )→ 0

arising from this graph is already quite complicated [MWZ1, BV], as thereare three different weights involved. The corresponding period functions when


the momentum p varies are given by elliptic dilogarithm functions [BV, ABW].There are generalizations to higher loop banana graphs [BKV].

In the literature, there are many more concrete examples of such periods, seethe work of Broadhurst-Kreimer [BK] and subsequent work. Besides multiplezeta values, there are for examples graphs G where the integral is related toperiods of K3 surfaces [BS].

15.3 Algebraic cycles and periods

In this section, we want to show how algebraic cycles in (higher) Chow groupsgive rise to Kontsevich-Zagier periods. Let us start with an example.

Example 15.3.1. Assume that k ⊂ C, and let X be a smooth, projective curveof genus g, and Z =

∑ki=1 aiZi ∈ CH1(X) be a non-trivial zero-cycle on X with

degree 0, i.e.,∑i ai = 0. Then we have a sequence of cohomology groups

0→ H1(Xan)→ H1(Xan\|Z|)→ H2|Z|(X

an) ∼=⊕i

Z(−1)Σ→H2(Xan,Z) ∼= Z(−1) .

The cycle Z defines a non-zero vector (a1, ..., ak) ∈⊕

i Z(−1) mapping to zeroin H2(Xan,Z). Hence, by pulling back, we obtain an extension

0→ H1(Xan)→ E → Z(−1)→ 0 .

The extension class of this sequence in the category of mixed Hodge structuresis known to be the Abel-Jacobi class of Z [C]. One can compute it in severalways. For example, one can choose a continuous chain γ with ∂γ =

∑i aiZi

and a basis ω1, ..., ωg of holomorphic 1-forms on Xan. Then the vector(∫γ

ω1, . . . ,

∫γ

ωg

)defines the Abel-Jacobi class in the Jacobian

Jac(X) =H1(Xan,C)

F 1H1(Xan,C) +H1(Xan,Z)∼=H0(Xan,Ω1

Xan)∨

H1(Xan,Z).

If X and the cycle Z are both defined over k, then obviously the Abel-Jacobiclass is defined by g period integrals in Peff(k). In the case of smooth, projectivecurves, the Abel-Jacobi map

AJ1 : CH1(X)hom → Jac(X)

gives an isomorphism when k = C.

15.3. ALGEBRAIC CYCLES AND PERIODS 301

One can generalize this construction to Chow groups. Let X be a smooth,projective variety over k ⊂ C, and Z ∈ CHq(X) a cycle which is homologousto zero. Then the Abel-Jacobi map

AJq : CHq(X)hom −→H2q−1(Xan,C)

F q +H2q−1(Xan,Z)∼= Ext1

MHS(Z(−q), H2q−1(Xan,Z)) ,

As in the example above, the cycle Z defines an extension of mixed Hodgestructures

0→ H2q−1(Xan)→ E → Z(−q)→ 0 ,

where E is a subquotient of H2q−1(Xan \ |Z|). The Abel-Jacobi class is givenby period integrals (∫

γ

ω1, ...,

∫γ

ωg

)in Griffiths’ Intermediate Jacobian

Jq(X) =H2q−1(Xan,C)

F qH2q−1(Xan,C) +H2q−1(Xan,Z)

∼=F qH2q−1(Xan,C)∨

H2q−1(Xan,Z).

Even more general, one may use Bloch’s higher Chow groups [Bl]. Higher Chowgroups are isomorphic to motivic cohomology in the smooth case by a result ofVoevodsky. In the general case, they only form a Borel-Moore homology theoryand not a cohomology theory [VSF]. Then the Abel-Jacobi map becomes

AJq,n : CHq(X,n)hom −→H2q−n−1(Xan,C)

F q +H2q−n−1(Xan,Z)∼= Ext1

MHS(Z(−q), H2q−n−1(Xan,Z)) ,

There are explicit formulas for AJq,n in [KLM, KLM2, Wei] on the level ofcomplexes. This shows that the higher Abel-Jacobi class is defined by periodintegrals which define numbers in Peff(k).

In analogy with the classical Chow groups, Spencer Bloch has found an explicitdescription of the extension of mixed Hodge structures associated to a cycleZ ∈ CHq(X,n)hom. This is explained in [DS, Scho2]. The periods associatedto this mixed Hodge structures can then be viewed as the periods associated toZ.

Let us describe this construction. We let n := (P1 \ 1)n. For varying n, thisdefines a cosimplicial object with face and degeneracy maps obtained by usingthe natural coordinate t on P1. Faces are given by setting ti = 0 or ti = ∞.By definition, a cycle Z in a higher Chow group CHq(X,n) is a subvariety ofX × n meeting all faces F = X × m ⊂ X × n for m < n properly, i.e., incodimension q. By looking at the normalized cycle complex, we may assumethat Z has zero intersection with all faces of X ×n. Removing the support ofZ, let U := X×n \ |Z|, and define ∂U to be the union of the intersection of U


with the codimension 1 faces of X × n. Then one obtains an exact sequence[DS, Lemma A.2]

0→ H2q−n−1(Xan)→ H2q−1(Uan, ∂Uan)→ H2q−1(Uan)→ H2q−1(∂Uan) ,

which can be pulled back to an extension E if Z is homologous to zero:

0→ H2q−n−1(Xan)→ E → Z(−q)→ 0 .

Hence, E is a subquotient of the mixed Hodge structure H2q−1(Uan, ∂Uan).This works for any cohomology satisfying certain axioms, see [DS]. In particular,applying it to singular or de Rham cohomology, we obtain an extension insidethe category of Nori motives.

For the category of Nori motives, extension groups are not known in general,and have only been computed in the situation of 1-motives [AB]. The extensiongroups of any abelian category MM(k) of mixed motives over k are conjecturallysupposed to be Adams eigenspaces of algebraic K-groups, or, equivalently, mo-tivic cohomology groups. For example, one expects that

Ext1MM(k)(Q(−q), H2q−n−q(X)) = H2q−n

M (X,Q(−q)) = Kn(X)(q)Q

for a smooth, projective variety X.

15.4 Periods of homotopy groups

In this section, we want to explain the periods associated to fundamental groupsand higher homotopy groups.

The topological fundamental group πtop1 (X(C), a) of an algebraic variety X

(defined over k ⊂ C) with base point a carries a MHS in the following sense.

First, look at the group algebra Qπtop1 (X(C), a), and the augmentation ideal

I := Ker(Qπtop1 (X, a)→ Q). Then the Malcev-type object

π1(X(C), a)Q := limn→∞

Q[πtop1 (X(C), a)]/In+1

should carry an Ind-MHS, as we will explain now. Beilinson observed that eachfinite step Qπtop

1 (X(C), a)/In+1 can be obtained as a MHS of a certain algebraicvariety defined over the same field k. This was known to experts for some time,and later worked out in [DG].

Theorem 15.4.1. Let M be any connected complex manifold and a ∈ M apoint. Then there is an isomorphism

Hn(M × · · · ×M︸︷︷︸n

, D;Q) ∼= Qπtop1 (M,a)/In+1,

and Hk(M × · · · ×M︸︷︷︸n

, D;Q) = 0 for k < n. Here D = ∪Di is a divisor,

where D0 = a × Mn−1, Dn+1 = Mn−1 × a, and, for 1 ≤ i ≤ n − 1,Di = M i−1 ×∆×Mn−i−1 with ∆ ⊂M ×M the diagonal.

15.4. PERIODS OF HOMOTOPY GROUPS 303

Proof. The proof in loc. cit., which we will not give here, proceeds by inductionon n, using the first projection p1 : Mn →M and the Leray spectral sequence.

In the framework of Nori motives, one can thus see that π1(X, a)Q immediatelycarries the structure of an Ind-Nori motive over k, since the Betti realization isobvious. Deligne-Goncharov [DG] and F. Brown [B2, B1] work instead withinthe framework of the abelian category of mixed Tate motives over Q of Levine[L2]. From this it follows, that π1(P1\0, 1,∞, a)Q is an Ind-mixed Tate motiveover Q (in fact, over Z as explained in [B1]). There is also a description of the deRham realization in [DG, B2, B1]. In particular, Brown showed that each MZVoccurs as a period of this Ind-MHS [B2, B1, D3], as we explained in Section 14.5.

Theorem 15.4.2. Every multiple zeta value occurs as a period of π1(P1(C) \0, 1,∞, a)Q. Furthermore, every multiple zeta value is a polynomial with Q-coefficients in multiple zate values with only 2 and 3 as entries.

Proof. See [B1, B2].

The proof of this theorem also implies that every mixed Tate motive over Zoccurs as a finite subquotient of the Ind-motive π1(P1 \ 0, 1,∞, a)Q.

Let us now look at higher homotopy groups πn(Xan) for n ≥ 2 of an algebraicvariety X over k ⊂ C. They carry a MHS rationally by a theorem of Morgan[Mo] and Hain [H]:

Theorem 15.4.3. The homotopy groups πn(Xan) ⊗ Q of a simply connectedand smooth projective variety over C carry a functorial mixed Hodge structurefor n ≥ 2.

This theorem has a natural extension to the non-compact case using logarithmicforms, and to the singular case using cubical hyperresolutions, see [PS] and [Na].

Example 15.4.4. Let X be a simply connected, smooth projective 3-fold overC. Then the MHS on π3(Xan)∨ is given by an extension

0→ H3(Xan,Q)→ Hom(π3(Xan),Q)→ Ker(S2H2(Xan,Q)→ H4(Xan,Q)

)→ 0

Carlson, Clemens, and Morgan [CCM] prove that this extension is given by theAbel-Jacobi class of a certain codimension 2 cycle Z ∈ CH2

hom(X), and theextension class of this MHS in the sense of [C] is given by the Abel-Jacobi class

AJ2(Z) ∈ J2(X) =H3(Xan,C)

F 2 +H3(Xan,Z).

The proof of Morgan uses the theory of Sullivan [Su]. In the simply connectedcase, there is a differential graded Lie algebra L(X,x) over Q, concentrated indegrees 0, −1, ..., such that

H∗(L(X,x)) ∼= π∗+1(Xan)⊗Q.


One can then use the cohomological description of L(X,x) and Deligne’s mixedHodge theory, to define the MHS on homotopy groups using a complex definedover k. We would like to mention that one can try to make this constructionmotivic in the Nori sense. At least for affine varieties, this was done in [Ga], seealso [CG, pg. 22]. In [G4], a description of periods of homotopy groups is givenin terms of Hodge correlators. This is not well understood yet.

From the approach in [Ga], one can see, at least in the affine case, that theperiods of the MHS on πn(Xan) are defined over k, i.e., are contained in Peff(k),when X is defined over k, since all motives involved in the construction aredefined over k.

15.5 Non-periods

The question whether a given transcendental complex number is a period num-ber in Peff(Q), i.e., is a Kontsevich-Zagier period, is very difficult to answer ingeneral, even though we know that there are only countably many of them. Forexample, we expect (but do not know) that the Euler number e is not a period.Also 1/π and Euler’s γ are presumably not effective periods, although no proofis known.

When Kontsevich-Zagier wrote their paper, the situation was like at the begin-ning of the 19th century for the study of algebraic and trancendental numbers.It took a lot of effort to prove that Liouville numbers

∑i 10−i!, e (Hermite) and

π (Lindemann) were transcendental.

In 2008, M. Yoshinaga [Y] first wrote down a non-period α = 0.77766444... in3-adic expansion

α =

∞∑i=1

εi3−i .

We will now explain how to define this number, and why it is not a period.First, we have to explain the notions of computable and elementary computablenumbers.

Computable numbers and equivalent notions of computable (i.e., equivalently,partial recursive) functions f : Nn0 → N0 were introduced by Turing [T], Kleeneand Church around 1936 following the ideas from Godel’s famous paper [G], seethe references in [Kl] . We refer to [Bri] for a modern treatment of such notionswhich is intended for mathematicians.

The modern theory of computable functions starts with the notion of certainclasses E of functions f : N0 → N0. For each class E there is then a notion ofE-computable real numbers. In the following definition we follow [Y], but thiswas defined much earlier, see for example [R, Spe].

Definition 15.5.1. A real number α > 0 is called E-computable, if there are

15.5. NON-PERIODS 305

sequences a(n), b(n), c(n) in E , such that∣∣∣∣ a(n)

b(n) + 1− α

∣∣∣∣ < 1

k, for all n ≥ c(k) .

The set of E-computable numbers, including 0 and closed under α 7→ −α, isdenoted by RE .

Some authors use the bound 2−k instead of 1k . This leads to an equivalent

notion only for classes E which contain exponentials n 7→ 2n.

If E = comp is the class of Turing computable [T], or equivalently Kleene’spartial recursive functions [Kl], then Rcomp is the set of computable real numbers.Computable complex numbers Ccomp are those complex numbers where the real-and imaginary part are computable reals.

Theorem 15.5.2. Rcomp is a countable subfield of R, and Ccomp = Rcomp(i) isalgebraically closed.

One can think of computable numbers as the set of all numbers that can beaccessed with a computer.

There are some important levels of hierarchies inside the set of computable reals

Rlow−elem ( Relem ( Rcomp ,

induced by the elementary functions of Kalmar (1943) [Ka], and the lower ele-mentary functions of Skolem (1962) [Sk]. There is also the related Grzegorczykhierarchy [Gr]. In order to define such hierarchies of real numbers, we will nowstudy functions f : Nn0 → N0 of several variables.

Definition 15.5.3. The class of lower-elementary functions is the smallest classof functions f : Nn0 → N0

• containing the zero-function, the successor function x 7→ x + 1 and theprojection function Pi : (x1, ..., xn) 7→ xi,

• containing the addition x + y, the multiplication x · y, and the modifiedsubtraction max(x− y, 0),

• closed under composition, and

• closed under bounded summation.

The class of elementary functions is the smallest class which is also closed underbounded products.

Here, bounded summation (resp. product) is defined as

g(x, x1, ..., xn) =∑a≤x

f(a, x1, ..., xn) resp.∏a≤x

f(a, x1, ..., xn) .


Elementary functions contain exponentials 2n, whereas lower elementary func-tion do not. The levels of the above hierarchy are strict [TZ].

The main result about periods proven in [Y, TZ] is:

Theorem 15.5.4. Real periods are lower elementary real numbers.

In fact, Yoshinaga proved that periods are elementary computable numbers,and Tent-Ziegler made the refinement that periods are even lower-elementarynumbers. The proofs are based on Hironaka’s theorem on semi-algebraic setswhich we have used already in chapter 2. The main idea is to reduce periods tovolumes of bounded semi-algebraic sets, and then use Riemann sums to approx-imate the volumes inside the class of lower elementary computable functions.

Corollary 15.5.5. One has inclusions:

Q ( Peff(Q) ⊂ Clow−elem ( Celem ( Ccomp .

Hence, in order to construct a non-period, one needs to exhibit a computablenumber which is not elementary computable. By Tent-Ziegler, it would alsobe enough to write down an elementary computable number which is not lowerelementary.

Here is how Yoshinaga proceeds. First, using a result of Mazzanti [Maz], one canshow that elementary functions are generated by composition from the followingfunctions:

• The successor function x 7→ x+ 1,

• the modified subtraction max(x− y, 0),

• the floor quotient (x, y) 7→ b xy+1c, and

• the exponential function (x, y) 7→ xy.

Using this, there is an explicit enumeration (fn)n∈N0of all elementary functions

f : N0 → N0. Together with the standard enumeration of Q>0, we obtain anexplicit enumeration (gn)n∈N0 of all elementary maps g : N0 → Q>0. Using atrick, see [Y, pg. 9], one can ”speed up” each function gn, so that gn(m) is aCauchy sequence (hence, convergent) in m for each n.

Following [Y], we therefore obtain

Relem = β0, β1, ..., where βn = limm→∞

gn(m) .

Finally, Yoshinaga defines

α := limn→∞

αn = limn→∞

n∑i=1

εi3−i ,

15.5. NON-PERIODS 307

where ε0 = 0, and recursively

εn+1 :=

0, if gn(n) > αn + 1

2·3n

1, if gn(n) ≤ αn + 12·3n

.

Now, it is quite easy to show that α does not occur in the list Relem = β0, β1, ...,see [Y, Prop. 17]. Note that the proof is essentially a version of Cantor’s diag-onal argument.


Part V

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